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19
Global entrainment of transcriptional systems to periodic inputs
, 2009
"... This paper addresses the problem of providing mathematical conditions that allow one to ensure that biological networks, such as transcriptional systems, can be globally entrained to external periodic inputs. Despite appearing obvious at first, this is by no means a generic property of nonlinear dyn ..."
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Cited by 26 (5 self)
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This paper addresses the problem of providing mathematical conditions that allow one to ensure that biological networks, such as transcriptional systems, can be globally entrained to external periodic inputs. Despite appearing obvious at first, this is by no means a generic property of nonlinear dynamical systems. Through the use of contraction theory, a powerful tool from dynamical systems theory, it is shown that certain systems driven by external periodic signals have the property that all their solutions converge to a fixed limit cycle. General results are proved, and the properties are verified in the specific cases of models of transcriptional systems as well as constructs of interest in synthetic biology. A selfcontained exposition of all needed results is given in the paper. Author Summary The activities of all living organisms are governed by complex sets of finely regulated biochemical reactions. Often, entrainment to certain external forcing signals helps control the timing and sequencing of reactions. For example, human activities are clearly regulated by the daynight cycle. That is, humans tend to adapt their function to some “external ” input. An important open problem is to understand the onset of entrainment and under what conditions it can be ensured in the presence of uncertainties, noise, and
A contractivity approach for probabilistic bisimulations of diffusion processes
 In Proceedings of the 48th IEEE Conference of Decision and Control
, 2009
"... Abstract — This work is concerned with the problem of characterizing and computing probabilistic bisimulations of diffusion processes. A probabilistic bisimulation relation between two such processes is defined through a bisimulation function, which induces an approximation metric on the expectatio ..."
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Cited by 8 (3 self)
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Abstract — This work is concerned with the problem of characterizing and computing probabilistic bisimulations of diffusion processes. A probabilistic bisimulation relation between two such processes is defined through a bisimulation function, which induces an approximation metric on the expectation of the (squared norm of the) distance between the two processes. We introduce sufficient conditions for the existence of a bisimulation function, based on the use of contractivity analysis for probabilistic systems. Furthermore, we show that the notion of stochastic contractivity is related to a probabilistic version of the concept of incremental stability. This relationship leads to a procedure that constructs a discrete approximation of a diffusion process. The procedure is based on the discretization of space and time. Given a diffusion process, we raise sufficient conditions for the existence of such an approximation, and show that it is probabilistically bisimilar to the original process, up to a certain approximation precision. I.
Finitetime Regional Verification of Stochastic Nonlinear Systems
"... Abstract—Recent trends pushing robots into unstructured environments with limited sensors have motivated considerable work on planning under uncertainty and stochastic optimal control, but these methods typically do not provide guaranteed performance. Here we consider the problem of bounding the pro ..."
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Cited by 7 (4 self)
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Abstract—Recent trends pushing robots into unstructured environments with limited sensors have motivated considerable work on planning under uncertainty and stochastic optimal control, but these methods typically do not provide guaranteed performance. Here we consider the problem of bounding the probability of failure (defined as leaving a finite region of state space) over a finite time for stochastic nonlinear systems with continuous state. Our approach searches for exponential barrier functions that provide bounds using a variant of the classical supermartingale result. We provide a relaxation of this search to a semidefinite program, yielding an efficient algorithm that provides rigorous upper bounds on the probability of failure for the original nonlinear system. We give a number of numerical examples in both discrete and continuous time that demonstrate the effectiveness of the approach. I.
Approximation Metrics based on Probabilistic Bisimulations for General StateSpace Markov Processes: a Survey
 HAS 2011
, 2011
"... This article provides a survey of approximation metrics for stochastic processes. We deal with Markovian processes in discrete time evolving on general state spaces, namely on domains with infinite cardinality and endowed with proper measurability and metric structures. The focus of this work is to ..."
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Cited by 3 (0 self)
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This article provides a survey of approximation metrics for stochastic processes. We deal with Markovian processes in discrete time evolving on general state spaces, namely on domains with infinite cardinality and endowed with proper measurability and metric structures. The focus of this work is to discuss approximation metrics between two such processes, based on the notion of probabilistic bisimulation: in particular we investigate metrics characterized by an approximate variant of this notion. We suggests that metrics between two processes can be introduced essentially in two distinct ways: the first employs the probabilistic conditional kernels underlying the two stochastic processes under study, and leverages notions derived from algebra, logic, or category theory; whereas the second looks at distances between trajectories of the two processes, and is based on the dynamical properties of the two processes (either their syntax, via the notion of bisimulation function; or their semantics, via sampling techniques). The survey moreover covers the problem of constructing formal approximations of stochastic processes according to the introduced metrics.
Analysis of discrete and hybrid stochastic systems by nonlinear contraction theory
 In Proceedings of the 10th International Conference on Control, Automation, Robotics and Vision (ICARCV 08
, 2008
"... AbstractWe investigate the stability properties of discrete and hybrid stochastic nonlinear dynamical systems. More precisely, we extend the stochastic contraction theorems (which were formulated for continuous systems) to the case of discrete and hybrid resetting systems. In particular, we show t ..."
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Cited by 2 (0 self)
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AbstractWe investigate the stability properties of discrete and hybrid stochastic nonlinear dynamical systems. More precisely, we extend the stochastic contraction theorems (which were formulated for continuous systems) to the case of discrete and hybrid resetting systems. In particular, we show that the mean square distance between any two trajectories of a discrete (or hybrid resetting) stochastic contracting system is upperbounded by a constant after exponential transients. Using these results, we study the synchronization of noisy nonlinear oscillators coupled by discrete interactions.
Synchronization and redundancy: Implications for robustness of neural learning and decision making
 Neural Computation
, 2011
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On three generalizations of contraction
 in Proc. 53rd IEEE Conf. on Decision and Control, LA, 2014. [Online]. Available: arXiv:1406.1474
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Synchronization can Control Regularization in Neural Systems via Correlated Noise Processes
"... To learn reliable rules that can generalize to novel situations, the brain must be capable of imposing some form of regularization. Here we suggest, through theoretical and computational arguments, that the combination of noise with synchronization provides a plausible mechanism for regularization i ..."
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Cited by 1 (1 self)
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To learn reliable rules that can generalize to novel situations, the brain must be capable of imposing some form of regularization. Here we suggest, through theoretical and computational arguments, that the combination of noise with synchronization provides a plausible mechanism for regularization in the nervous system. The functional role of regularization is considered in a general context in which coupled computational systems receive inputs corrupted by correlated noise. Noise on the inputs is shown to impose regularization, and when synchronization upstream induces timevarying correlations across noise variables, the degree of regularization can be calibrated over time. The resulting qualitative behavior matches experimental data from visual cortex. 1
Tradeoffs
"... between retroactivity and noise in connected transcriptional components ..."
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