Singular perturbation analysis and gene regulatory networks with delay

Shlykova, Irina; Ponosov, Arcady

doi:10.1016/j.na.2010.01.016

Cited by 8 publications

(15 citation statements)

References 8 publications

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“…The analysis around such points, also known as 'singular steady states', was initiated in [12] and developed in later works (see e.g. [3], [7], [13], [15]). Assumption 5 is of technical yet generic character; see [8] for more details.…”

Section: Singular Perturbation Analysis In Codimensionmentioning

confidence: 99%

“…1 , x q R ) of the full initial problem (13) and the solution (z * 1 , x * R ) of the reduced equationṡ…”

Section: Theorem 3 If Assumptions 5-7 Are Fulfilled Then the Solutimentioning

confidence: 99%

“…In this framework the parameters of the model usually have a biochemical or physical meaning; for details see [ [10], where the Hill functional form is derived from the analysis of the equilibrium binding of the transcription factor to its site on the promoter. Calculations with the Hill functions are well-elaborated, in particular, in the analysis of sliding modes [8], [13].…”

Section: Examplementioning

confidence: 99%

“…[3], [6]), while the other utilizes singular perturbation analysis (see e.g. [8], [13], [15]), where smooth response functions of sigmoid-type replace the step functions.…”

Section: Introductionmentioning

confidence: 99%

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Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

Tafintseva¹,

Machina²,

Ponosov³

2013

Nonlinear Analysis: Real World Applications

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We describe generic sliding modes of piecewise-linear systems of differential equations arising in the theory of gene regulatory networks with Boolean interactions. We do not make any a priori assumptions on regulatory functions in the network and try to understand what mathematical consequences this may have in regard to the limit dynamics of the system. Further, we provide a complete classification of such systems in terms of polynomial representations for the cases where the discontinuity set of the right-hand side of the system has a codimension 1 in the phase space. In particular, we prove that the multilinear representation of the underlying Boolean structure of a continuous-time gene regulatory network is only generic in the absence of sliding trajectories. Our results also explain why the Boolean structure of interactions is too coarse and usually gives rise to several non-equivalent models with smooth interactions.

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Section: Singular Perturbation Analysis In Codimensionmentioning

confidence: 99%

“…1 , x q R ) of the full initial problem (13) and the solution (z * 1 , x * R ) of the reduced equationṡ…”

Section: Theorem 3 If Assumptions 5-7 Are Fulfilled Then the Solutimentioning

confidence: 99%

Section: Examplementioning

confidence: 99%

“…[3], [6]), while the other utilizes singular perturbation analysis (see e.g. [8], [13], [15]), where smooth response functions of sigmoid-type replace the step functions.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

Tafintseva¹,

Machina²,

Ponosov³

2013

Nonlinear Analysis: Real World Applications

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“…Rate equation models also occur in other branches of mathematical biology, for example in the study of gene regulatory networks [13,14,15,16,17,18,19]. A simple gene regulatory network model is given as…”

Section: Introductionmentioning

confidence: 99%

Neural firing rate model with a steep firing rate function

Yousaf

Ponosov

Wyller

et al. 2013

Nonlinear Analysis: Real World Applications

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In this study we justify rigorously the approximation of the steep firing rate functions with a unit step function in a two-population neural firing rate model with steep firing rate functions. We do this justification by exploiting the theory of switching dynamical systems. It has been demonstrated that switching dynamics offer a possibility of simplifying the dynamical system and getting approximations of the solution of the system for any specific choice of parameters. In this approach the phase space of the system is divided into regular and singular domains, where the limit dynamics can be written down explicitly. The advantages of this method are illustrated by number of numerical examples for different cases of the singular domains (i.e. for black, white and transparent walls) and for specific choices of parameters involved. General conditions have been formulated on these parameters to give black, white and transparent walls. Further, the existence and stability of regular and singular stationary points have been investigated. It has been shown that the regular stationary points (i.e. stationary points inside the regular domains) are always stable and this property is preserved for smooth and sufficiently steep activation functions. In the most technical part of the paper we have provided conditions on the existence and stability of singular stationary points (i.e. those belonging to the singular domains). For the * Corresponding Author: Tel. +47 64965489; fax.+47 64965401.Email existence results, the implicit function theorem has been used, whereas the stability of singular stationary points is addressed by applying the singular perturbation analysis and the Tikhonov theorem.

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Infinite Dimensions

Kuehn¹

2014

Applied Mathematical Sciences

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Singular perturbation analysis and gene regulatory networks with delay

Cited by 8 publications

References 8 publications

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

Polynomial representations of piecewise-linear differential equations arising from gene regulatory networks

Neural firing rate model with a steep firing rate function

Infinite Dimensions

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