On limit cycles of monotone functions with symmetric connection graph

Aracena, Julio; Demongeot, Jacques; Goles, Éric

doi:10.1016/j.tcs.2004.03.010

Cited by 40 publications

(36 citation statements)

References 4 publications

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“…Under these hypotheses, the authors of [45,73,74] proved the following results: Theorem 1. If all cycles of the undirected version of G are positive then there exists a vector x = (x 1 , .…”

Section: Appendix B Biological Regulation Network Modelingmentioning

confidence: 97%

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“Immunetworks”, intersecting circuits and dynamics

Demongeot

Elena

Noual

et al. 2011

Journal of Theoretical Biology

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This paper proposes a study of biological regulatory networks based on a multi-level strategy. Given a network, the first structural level of this strategy consists in analysing the architecture of the network interactions in order to describe it. The second dynamical level consists in relating the patterns found in the architecture to the possible dynamical behaviours of the network. It is known that circuits are the patterns that play the most important part in the dynamics of a network in the sense that they are responsible for the diversity of its asymptotic behaviours. Here, we pursue further this idea and argue that beyond the influence of underlying circuits, intersections of circuits also impact significantly on the dynamics of a network and thus need to be payed special attention to. For some genetic regulation networks involved in the control of the immune system ("immunetworks"), we show that the small number of attractors can be explained by the presence, in the underlying structures of these networks, of intersecting circuits that "inter-lock".

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Section: Appendix B Biological Regulation Network Modelingmentioning

confidence: 97%

“…First, as it has been explained in Appendix D, in certain applications [27,45,67,73,74], the state 0 is replaced by −1 by a simple change of variable.…”

Section: Appendix E Relationships Between the Different Formalismsmentioning

confidence: 99%

“Immunetworks”, intersecting circuits and dynamics

Demongeot

Elena

Noual

et al. 2011

Journal of Theoretical Biology

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“…The mechanistic basis for this likely lies in the fact that negative feedback loops have been conjectured (and subsequently shown) to be essential for periodic behavior--that is, they tend to keep a system on periodic attractors [32,34,38]. While this result has been confirmed in Boolean networks [39][40][41][42], Boolean modeling has also been used recently to show that as the number of feedback loops (particularly independent negative ones) increases in a network, the cyclic attractors of a network tend to become longer and the dynamics are much closer to chaotic [43]. Using a novel measure of independent negative feedback loops called "distanceto-positive-feedback," it was shown that, as the number of independent negative feedback loops increases, there tends to be a smaller number of larger cycles; a cycle structure associated with chaotic dynamics.…”

Section: Mechanisms Of Dynamics; Boolean Studies Of the Feedback Loopmentioning

confidence: 97%

“…This function is a crucial property of memory, hence it can be argued that one of the main functions of positive feedback is to form the basis for cellular "memory modules" [32,45,46]. The dependence of multistability on positive feedback loops has also been demonstrated in Boolean models [38][39][40][41]47].…”

Section: Mechanisms Of Dynamics; Boolean Studies Of the Feedback Loopmentioning

confidence: 99%

Boolean Modeling of Biochemical Networks

Helikar¹,

Kochi

Konvalina

et al. 2011

TOBIOIJ

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Abstract:The use of modeling to observe and analyze the mechanisms of complex biochemical network function is becoming an important methodological tool in the systems biology era. Number of different approaches to model these networks have been utilized--they range from analysis of static connection graphs to dynamical models based on kinetic interaction data. Dynamical models have a distinct appeal in that they make it possible to observe these networks in action, but they also pose a distinct challenge in that they require detailed information describing how the individual components of these networks interact in living cells. Because this level of detail is generally not known, dynamic modeling requires simplifying assumptions in order to make it practical. In this review Boolean modeling will be discussed, a modeling method that depends on the simplifying assumption that all elements of a network exist only in one of two states.Keywords: Systems biology, dynamical modeling, signal transduction, boolean networks.The modern era of cellular biochemistry has been characterized by the discovery of large interaction networks that are notable for their highly nonlinear connection structures. Such structures complicate study since they are often quite difficult to reduce into linear subsets that function in isolation from the embedding network [1,2]. It is for this reason that the concept of "systems biology" has been developed and is now being applied to these complicated biochemical networks [1,3,4].One of the first steps taken toward applying a systems approach to understanding biochemical networks was to begin organizing the connection information determined in the laboratory by creating connection maps (wiring diagrams) that could then be analyzed using graph theory concepts. While these studies have been highly successful [5], the fact that cells are dynamical systems means the next major step in the study of these systems involves the creation of models that take into account how they move and function in time. Thus we are now in a new era of creating large-scale dynamical models of biochemical systems.An obvious choice for the modeling of dynamical systems is to describe the system with a system of differential equations, and this approach has been used extensively to study protein and/or gene interactions in a variety of relatively small models (for examples, see [6][7][8][9]). One of the advantages of continuous modeling via differential equations is that one can potentially describe molecular interactions with high precision and in quantitative terms that correspond to realistic laboratory measurements. However, that precision tends to make them computationally unwieldy for the large scale models necessary for the systems approach. There *Address correspondence to this author at the Department of Mathematics, University of Nebraska at Omaha, 6001 Dodge Street, Omaha, NE 68182, USA; Tel: 402 554-3110: Fax: 402 554-2975; E-mail: jrogers@unmc.edu is currently a great deal of work being done in order to simpl...

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“…Note that the interaction graph contains only one connected component having at least one (here two) positive circuit of interactions (a circuit is positive if its number of inhibiting edges is even). Hence, from [16,17,18,19,20,21,22], we can expect only 2 1 = 2 fixed configurations for the network dynamics and an upper bound for this number of 2 2 . On Table 3, we see that, if the state of p27 and miRNA 159 are not fixed to particular values, then this number is in reality 2, plus one (resp.…”

Section: Cell Cyclementioning

confidence: 97%

Structural Sensitivity of Neural and Genetic Networks

Amor

Demongeot

Sené

2008

MICAI 2008: Advances in Artificial Intelligence

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Abstract. This paper aims at giving new results on the structural sensitivity of biological networks represented by threshold Boolean networks and ruled by Hopfield-like evolution laws classically used in the context of neural and genetic networks. Indeed, the objective is to present how certain changes and/or perturbations in such networks can modify significantly their asymptotic behaviour. More precisely, this work has been focused on three different kinds of what we think to be relevant in the biological area of robustness (in both theoretical and applied frameworks): the boundary sensitivity (external fields, hormone flows, ...), the state sensitivity (axonal or somatic modulations, microRNAs actions, ...) and the updating sensitivity.

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On limit cycles of monotone functions with symmetric connection graph

Cited by 40 publications

References 4 publications

“Immunetworks”, intersecting circuits and dynamics

“Immunetworks”, intersecting circuits and dynamics

Boolean Modeling of Biochemical Networks

Structural Sensitivity of Neural and Genetic Networks

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