Stability switching and Hopf bifurcation in a multiple-delayed neural network with distributed delay

Ncube, Israel

doi:10.1016/j.jmaa.2013.05.021

Cited by 26 publications

(13 citation statements)

References 4 publications

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“…Taken together, these two results suggest that the promoter architecture of the l system may have evolved to allow for both robust bistability due to the positive feedback as well as reduced variability or other benefit of the small negative autoregulation. Although relatively little known within the biological sciences, Sturm's theorem has found applicability in a number of other areas where polynomials play an important role, including computational mathematics [25], dynamical systems [26,27], robotics [28] and finance [29]. Additional biological applications are possible, for example, as a tool to predict a large number of new bistable topologies or rule out those that do not have the capacity for bistability (such as chemical reaction network theory, previously [13,24]).…”

Section: Discussionmentioning

confidence: 99%

“…Although relatively little known within the biological sciences, Sturm's theorem has found applicability in a number of other areas where polynomials play an important role, including computational mathematics [25], dynamical systems [26,27], robotics [28] and finance [29]. Additional biological applications are possible, for example, as a tool to predict a large number of new bistable topologies or rule out those that do not have the capacity for bistability (such as chemical reaction network theory, previously [13,24]).…”

Section: Discussionmentioning

confidence: 99%

See 1 more Smart Citation

An analytical approach to bistable biological circuit discrimination using real algebraic geometry

Siegal-Gaskins

Franco

Zhou

et al. 2015

J. R. Soc. Interface.

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Biomolecular circuits with two distinct and stable steady states have been identified as essential components in a wide range of biological networks, with a variety of mechanisms and topologies giving rise to their important bistable property. Understanding the differences between circuit implementations is an important question, particularly for the synthetic biologist faced with determining which bistable circuit design out of many is best for their specific application. In this work we explore the applicability of Sturm's theorem-a tool from nineteenth-century real algebraic geometryto comparing 'functionally equivalent' bistable circuits without the need for numerical simulation. We first consider two genetic toggle variants and two different positive feedback circuits, and show how specific topological properties present in each type of circuit can serve to increase the size of the regions of parameter space in which they function as switches. We then demonstrate that a single competitive monomeric activator added to a purely monomeric (and otherwise monostable) mutual repressor circuit is sufficient for bistability. Finally, we compare our approach with the Routh-Hurwitz method and derive consistent, yet more powerful, parametric conditions. The predictive power and ease of use of Sturm's theorem demonstrated in this work suggest that algebraic geometric techniques may be underused in biomolecular circuit analysis.

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Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

An analytical approach to bistable biological circuit discrimination using real algebraic geometry

Siegal-Gaskins

Franco

Zhou

et al. 2015

J. R. Soc. Interface.

View full text Add to dashboard Cite

show abstract

“…Though relatively little-known within the biological sciences, Sturm's theorem has found applicability in a number of other areas where polynomials play an important role, including computational mathematics [25], dynamical systems [26,27], robotics [28], and finance [29]. Additional biological applications are possible, for example as a tool to predict a large number of new bistable topologies or rule out those that do not have the capacity for bistability (like Chemical Reaction Network Theory, previously [13,24]).…”

Section: Discussionmentioning

confidence: 99%

An analytical approach to bistable biological circuit discrimination using real algebraic geometry

Siegal-Gaskins

Franco

Zhou

et al. 2014

Preprint

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Biomolecular circuits with two distinct and stable steady states have been identified as essential components in a wide range of biological networks, with a variety of mechanisms and topologies giving rise to their important bistable property. Understanding the differences between circuit implementations is an important question, particularly for the synthetic biologist faced with determining which bistable circuit design out of many is best for their specific application. In this work we explore the applicability of Sturm's theorem-a tool from 19th-century real algebraic geometry-to comparing "functionally equivalent" bistable circuits without the need for numerical simulation. We first consider two genetic toggle variants and two different positive feedback circuits, and show how specific topological properties present in each type of circuit can serve to increase the size of the regions of parameter space in which they function as switches. We then demonstrate that a single competitive monomeric activator added to a purely-monomeric (and otherwise monostable) mutual repressor circuit is sufficient for bistability. Finally, we compare our approach with the Routh-Hurwitz method and derive consistent, yet more powerful, parametric conditions. The predictive power and ease of use of Sturm's theorem demonstrated in this work suggests that algebraic geometric techniques may be underutilized in biomolecular circuit analysis.

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“…Majee and Roy [20] dealt with the temporal dynamics of two-neuron continuous network model with time delay. For more work on this aspect, one can see [2,[21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39]. In 2008, Gupta et al [40] considered the following three neurons network with distributed delay…”

Section: Introductionmentioning

confidence: 99%

Bifurcation Analysis in a Three-Neuron Artificial Neural Network Model with Distributed Delays

Zhang

2015

Neural Process Lett

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In this paper, a three-neuron artificial neural network model with distributed delays is considered. Its dynamics is investigated in term of the linear stability analysis and Hopf bifurcation analysis. By regarding the sum of two delays as a bifurcation parameter and analyzing the associated characteristic equation, we find that Hopf bifurcation occurs when the bifurcation parameter passes through some certain values. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions are derived by using the normal form method and center manifold theory. Finally, computer simulations are given to support the theoretical predictions.

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Stability switching and Hopf bifurcation in a multiple-delayed neural network with distributed delay

Cited by 26 publications

References 4 publications

An analytical approach to bistable biological circuit discrimination using real algebraic geometry

An analytical approach to bistable biological circuit discrimination using real algebraic geometry

An analytical approach to bistable biological circuit discrimination using real algebraic geometry

Bifurcation Analysis in a Three-Neuron Artificial Neural Network Model with Distributed Delays

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