Invariance Conditions for Nonlinear Dynamical Systems

Horváth, Zoltán; Song, Yunfei; Terlaky, Tamás

doi:10.1007/978-3-319-42056-1_8

Cited by 8 publications

(4 citation statements)

References 22 publications

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“…While these studies provided interesting insight into the mechanisms underlying MDD in the two sexes, prior studies have not consistently compared male and female profiles and, as such, the extent to which the transcriptional signatures defining MDD differs in males vs. females remains unknown. Leveraging the intrinsic properties of gene coexpression network analysis 78–81 , we highlighted the fundamental differences in stress responses in males vs. females and provided a framework for better understanding the molecular basis of the sexual dimorphism characterizing MDD. Our findings suggest that MDD in males and females may arise in part from the actions of similar gene modules, which share cellular and biological specificity, but which are organized and expressed differently across brain regions in the two sexes.…”

Section: Discussionmentioning

confidence: 99%

Sex-specific transcriptional signatures in human depression

Labonté

Engmann

Purushothaman

et al. 2017

Nat Med

581

593

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Summary Major depressive disorder (MDD) is a leading cause of disease burden worldwide. While the incidence, symptoms and treatment of MDD all point toward major sex differences, the molecular mechanisms underlying this sexual dimorphism remain largely unknown. Here, combining differential expression and weighted gene coexpression network analyses, we provide a comprehensive characterization of male and female transcriptional profiles associated with MDD across 6 brain regions. We overlap our human profiles with those from a mouse model of chronic variable stress and capitalize on converging pathways to define molecular and physiological mechanisms underlying the expression of stress susceptibility in males and females. Our results show a major rearrangement of transcriptional patterns, with male and female transcriptional profiles sharing very limited overlap, an effect seen in depressed humans and in stressed mice. We identify male and female hub genes and confirm their sex-specific impact as stress-susceptibility mediators. For example, downregulation of the female-specific hub gene DUSP6 in prefrontal cortex mimics stress susceptibility in females only by increasing ERK signaling and pyramidal neuron excitability. Such DUSP6 downregulation also recapitulates the transcriptional remodelling that occurs in PFC of depressed females. Together, our findings reveal dramatic sexual dimorphism at the transcriptional level in MDD and highlight the importance of studying sex-specific treatments for this disorder.

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Section: Discussionmentioning

confidence: 99%

Sex-specific transcriptional signatures in human depression

Labonté

Engmann

Purushothaman

et al. 2017

Nat Med

581

593

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“…We show that if the variational system associated with the nonlinear system is k-positive then the nonlinear system admits certain invariant sets that can be described explicitly. Invariant sets play a significant role in many controltheoretic and engineering applications (see e.g., the survey [3] and the more recent PhD thesis [29]), yet analytic verification that a set is invariant is a non-trivial problem [10]. We also show that 2-cooperative systems satisfy a Poincaré-Bendixon property.…”

Section: Introductionmentioning

confidence: 71%

A Generalization of Linear Positive Systems with Applications to Nonlinear Systems: Invariant Sets and the Poincaré-Bendixson Property

Weiss,

Margaliot

2019

Preprint

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The dynamics of linear positive systems maps the positive orthant to itself. In other words, it maps a set of vectors with zero sign variations to itself. This raises the following question: what linear systems map the set of vectors with k sign variations to itself? We address this question using tools from the theory of cooperative dynamical systems and the theory of totally positive matrices. This yields a generalization of positive linear systems called k-positive linear systems, that reduces to positive systems for k = 1. We describe applications of this new type of systems to the analysis of nonlinear dynamical systems. In particular, we show that such systems admit certain explicit invariant sets, and for the case k = 2 establish the Poincaré-Bendixon property for certain trajectories.

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Section: Introductionmentioning

confidence: 99%

“…For a good survey of the art, refer to the survey paper of F. Blanchini [4]. An extension of positive invariance conditions to nonlinear dynamical systems and stochastic systems can be found in [5,6]. The necessary and sufficient conditions of positively invariant sets for linear dynamical systems are studied in this paper.…”

Section: Introductionmentioning

confidence: 99%

Bulletin of the Polish Academy of Sciences: Technical Sciences

Yang,

2020

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Positively invariant sets play an important role in the theory and applications of dynamical systems. The stability in Lyapunov sense of equilibrium x = 0 is equivalent to the existence of the ellipsoidal positively invariant sets. The constraints on the state and control vectors of dynamical systems can be formulated as polyhedral positively invariant sets in practical engineering problems. Numerical checking method of positive invariance of polyhedral sets is addressed in this paper. The validation of the positively invariant sets can be done by solving LPs which can be easily implemented numerically. The effectiveness of the proposed checking method is illustrated by examples. Compared with the now existing algebraic methods, numerical checking method is attractive and, importantly, easy to be implemented.

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Invariance Conditions for Nonlinear Dynamical Systems

Cited by 8 publications

References 22 publications

Sex-specific transcriptional signatures in human depression

Sex-specific transcriptional signatures in human depression

A Generalization of Linear Positive Systems with Applications to Nonlinear Systems: Invariant Sets and the Poincaré-Bendixson Property

Bulletin of the Polish Academy of Sciences: Technical Sciences

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