Delay-Dependent Stability for Genetic Regulatory Networks

From biochemical reaction principles, a genetic regulatory network can be described by a group of nonlinear differential equations with time delays. Previous studies have investigated delay-independent stability of genetic regulatory networks with time delays. However, if it is delay-independently stable, a genetic regulatory network loses other interesting properties such as oscillation. In this paper, we provide a computational method for computing the maximal delay interval over which the genetic regulatory network maintains stability, and beyond which the network will not be stable. Furthermore we prove that when its delay is exactly the maximal delay the network is oscillated. In addition, the formula for calculating the oscillation period is presented. The autoregulatory genetic network in zebrafish is used as an example to illustrate the presented method. The oscillation period calculated from our method is very close to that observed from the real-life of a zebrafish, which indicates the effectiveness of our method.

Although Ordinary Differential Equations (ODEs) have been used to model Genetic Regulatory Networks (GRNs) in many previous works, their steady-state behaviors are not well studied. However, a phenotype corresponds to a steady-state gene expression pattern and steady-state analysis of GRNs can provide valuable information on the stability of the GRNs, insights into cellular regulatory mechanisms underlying disease development as well as possible interventions for disease control. In this study, the steady-state behaviors of the nonlinear GRN models are analyzed based on time series data. The steady-state solutions and stability of nonlinear GRNs including polynomial model, sigmoidal model and S-system model are discussed in details.

Robust stability for genetic regulatory networks with linear fractional uncertainties

A reduced-order approach to analyze stability of genetic regulatory networks with discrete time delays

This paper presents a analysis approach to asymptotic stability of the genetic regulatory networks with interval time-varying delays, the delays play an important role in dynamics of genetic networks and cannot be ignored in the modeling of genetic regulation due to slow biochemical reactions. The free weighting matrices are employed to deal with cross product items, and the convexity of the matrix function is fully utilized in our proof to obtain less conservativeness, by utilizing some techniques, both the upper and lower bounds of the delays are brought into final stability conditions. Based on the Lyapunov stability theory and linear matrix inequality, sufficient conditions are given to ensure the asymptotic stability of the genetic regulatory networks, the obtained conditions are derived in terms of LMIs which are easy to be verified via the LMI toolbox. Illustrative examples are provided to show the effectiveness of our proposed method.

Asymptotic stability for neural networks with mixed time-delays: The discrete-time case

In this paper, we investigate the robust asymptotic stability problem of genetic regulatory networks with time-varying delays and polytopic parameter uncertainties. Both cases of differentiable and nondifferentiable time-delays are considered, and the convex polytopic description is utilized to characterize the genetic network model uncertainties. By using a Lyapunov functional approach and linear matrix inequality (LMI) techniques, the stability criteria for the uncertain delayed genetic networks are established in the form of LMIs, which can be readily verified by using standard numerical software. An important feature of the results reported here is that all the stability conditions are dependent on the upper and lower bounds of the delays, which is made possible by using up-to-date techniques for achieving delay dependence. Another feature of the results lies in that a novel Lyapunov functional dependent on the uncertain parameters is utilized, which renders the results to be potentially less conservative than those obtained via a fixed Lyapunov functional for the entire uncertainty domain. A genetic network example is employed to illustrate the applicability and usefulness of the developed theoretical results.

In this paper, the robust stability analysis of a problem is investigated for a class of discrete recurrent neural networks with distributed time varying delays for delay dependent case. The problem is to determine the robust stability by employing Lyapunov–Krasovskii stability theory. The class of neural network under some consideration is globally asymptotically stable if the quadratic matrix inequality involving several parameters is less than zero. Furthermore, a Linear Matrix Inequality (LMI) approach is provided to show the stability analysis. The numerical examples are given to show the usefulness of the proposed robust stability conditions. The numerical simulation is proved using MATLAB.

Global exponential stability for uncertain cellular neural networks with multiple time-varying delays via LMI approach

This brief investigates the problem of global exponential stability analysis for discrete recurrent neural networks with time-varying delays. In terms of linear matrix inequality (LMI) approach, a novel delay-dependent stability criterion is established for the considered recurrent neural networks via a new Lyapunov function. The obtained condition has less conservativeness and less number of variables than the existing ones. Numerical example is given to demonstrate the effectiveness of the proposed method.

This paper deals with the global asymptotic stability problem for Hopfield neural networks with time-varying delays. By resorting to the integral inequality and constructing a Lyapunov-Krasovskii functional, a novel delay-dependent condition is established to guarantee the existence and global asymptotic stability of the unique equilibrium point for a given delayed Hopfield neural network. This criterion is expressed in terms of linear matrix inequalities (LMIs), which can be easily checked by utilizing the recently developed algorithms for solving LMIs. Examples are provided to demonstrate the effectiveness and reduced conservatism of the proposed condition.

In this note, the global asymptotic stability of a class of delayed cellular neural networks is studied. Some new sufficient conditions are presented for the uniqueness of equilibrium point and the global stability of cellular neural networks with time varying delay by constructing Lyapunov functional and using linear matrix inequality (LMI). A numerical example is presented to illustrate the effectiveness of our theoretical results.

A robust delay-dependent guaranteed cost PID multivariable output feedback controller design for time-varying delayed systems: An LMI optimization approach

This paper is concerned with the problem of delay-derivative-dependent stability analysis for Markovian jumping neutral-type interval time-varying delay systems with mixed delays. The first, based on the Lyapunov-Krasovskii stability theory for functional differential equations and the linear matrix inequality approach, a delay-range-dependent condition for neutral-type Markovian jumping systems (NMJSs) with time-varying delays is obtained, which can guarantee global asymptotical stability of these NMJSs. Unlike the previous methods, the upper bound of the discrete delay and neutral delay derivative is taken into consideration even if this upper bound is larger than or equal to $$1(\dot{h}(t)<1,\dot{\tau}(t)<1)$$ . It is proved that the obtained results are less conservative than the existing ones. In our result, the time-varying delays are only assumed to be bounded. This has undoubtedly extended its application range. To better handle the problem on stability for neutral control systems, in which time-varying delay was involved, a stability criterion with less conservatism was put forward. Moreover, because our results in this paper are all based on the linear matrix inequality (LMI) approach, we can utilize Matlab’s LMI Control Toolbox to verify the global stability of correlation systems conveniently. As far as we are aware, this paper seems to be the first to discuss stability problems for neutral-type Markovian jumping systems with delay-derivative-dependence and delay-range-dependence.

The effect of parameter uncertainties on stability of delayed genetic regulatory networks is investigated. With the presence of parameter uncertainties, the delayed coupled genetic regulatory network is a parameter-dependent non-linear system, in which not only the dynamics are indefinite but also its equilibrium point is unknown. The authors first derive a sufficient condition to guarantee robust stability of uncertain genetic regulatory networks by introducing convergence region. Then, the estimation of convergence region bound is obtained by using Lyapunov stability theory and linear matrix inequality. It is worth noting that these results are derived in the case that the equilibrium point of parameter-dependent non-linear system is unknown. Finally, two examples are given to illustrate the effectiveness of the proposed results.