Elsevier

Neural Networks

Volume 22, Issues 5–6, July–August 2009, Pages 658-663
Neural Networks

2009 Special Issue
Global stability analysis and robust design of multi-time-scale biological networks under parametric uncertainties

https://doi.org/10.1016/j.neunet.2009.06.051Get rights and content

Abstract

Biological networks are prone to internal parametric fluctuations and external noises. Robustness represents a crucial property of these networks, which militates the effects of internal fluctuations and external noises. In this paper biological networks are formulated as coupled nonlinear differential systems operating at different time-scales under vanishing perturbations. In contrast to previous work viewing biological parametric uncertain systems as perturbations to a known nominal linear system, the perturbed biological system is modeled as nonlinear perturbations to a known nonlinear idealized system and is represented by two time-scales (subsystems). In addition, conditions for the existence of a global uniform attractor of the perturbed biological system are presented. By using an appropriate Lyapunov function for the coupled system, a maximal upper bound for the fast time-scale associated with the fast state is derived. The proposed robust system design principles are potentially applicable to robust biosynthetic network design. Finally, two examples of two important biological networks, a neural network and a gene regulatory network, are presented to illustrate the applicability of the developed theoretical framework.

Introduction

Both gene regulatory and neural networks, combining coupled dynamics of fast and slow states, constitute an important class of biological networks. The in silico implementation process of such networks is very sensitive to parameter perturbations. Errors in parameters, such as perturbations of feedback gains of activation functions or concentrations of chemical species, interconnection weights, and external inputs, are caused by data inaccuracies or computation errors. These perturbations can lead to location errors in equilibria, to instabilities, and even to spurious states. Therefore, a rigorous understanding of the qualitative robustness properties of biological systems with respect to parameter variations on both fast or slow time-scales becomes imperative. In this paper, biological networks, as coupled nonlinear differential systems operating at different time-scales under vanishing perturbations, are formulated. In previous work (Meyer-Baese, Roberts, & Yu, 2007), biological parametric uncertain systems have been considered as a known nominal linear system with perturbations (Sen and Datta, 1993, Shao, 2004), but in this paper the perturbed biological system is modeled as nonlinear perturbations to a known nonlinear idealized system and is represented by two time-scales (subsystems).

There is no research reported on the stability of two-time-scale biological networks modeling both fast and slow nonlinear dynamics with nonlinear uncertainties. However, many approaches based on neural network techniques have been proposed for various nonlinear systems. For example, the stabilization tracking problem for MIMO nonlinear systems has been also studied in recent years including nonlinear identification and control using recurrent neural networks (for example see Sanchez, Loukianov, and Felix (2003) and references therein), a network identifier in block form (Kosmatopoulos, Christodoulou, & Ioannou, 1997) and adaptive output control design (Hoseini, Farrokhi, & Koshkouei, 2009). In particular, because of approximation errors inherent in neural networks, when the number of neurons is limited or initialization of weights is not suitable, most of these methods can guarantee only uniformly ultimately bounded stability. A robust adaptive output feedback control design method for uncertain nonlinear systems which does not rely on state estimation has been proposed in Hoseini, Farrokhi, and Koshkouei (2008). This approach is applicable to systems with unknown but bounded dimensions and with known relative degree, and a neural network is considered which approximates and adaptively compensates for unknown plant nonlinearities.

In this paper, the robustness properties of biological networks, modeled by a system of competitive differential equations, are studied from a rigorous analytic standpoint and the theoretical results are applied to nonlinear uncertain singularly perturbed systems (Binning & Goodall, 2000). The networks under study model the nonlinear dynamics of both fast and slow states under consideration of nonlinear uncertainties.

The developed theory is applied to two important cases of biological networks: a multi-time-scale recurrent neural network and gene regulatory networks describing the heat shock response in Escherichia coli and the tryptophan regulatory network. Sufficient stability conditions for these two types of network are derived.

In the following, some notations which are used in this paper are introduced.

Notations:

    λmax(P):

    the maximum eigenvalue of a symmetric matrix P.

    λmin(P):

    the minimum eigenvalue of a symmetric matrix P.

    A:

    (λmax(ATA))12 for a given matrix A.

    ,:

    inner product on RN.

    :

    Euclidean norm.

    (v)(x):

    gradient field vx(x).

    C(RN):

    the space of all continuous functions mapping RNRN.

    C1(RN):

    the space of continuous functions which have continuous first-order partial derivatives. Note that C1(RN)C(RN).

    Df(x):

    Fréchet derivative of f.

    RN×N:

    the set of real square matrices of order N×N.

    Lfv:

    RNR or (v)(x),f(x): Lie derivative of a scalar field xv(x)R along the vector field fRN.

Section snippets

Problem statement

The class of perturbed uncertain biological systems considered is a two-time-scale system consisting of two coupled subsystems with the following structure (see, for example, Binning and Goodall (2000)): Ṡ(t)=a(t,S(t),x(t))ϵẋ(t)=b(t,S(t),x(t),ϵ) where a(t,S,x)=f1(x)S+h1(t,S,x)b(t,S,x,ϵ)=A(t)[S+f2(x)]+h2(t,S,x,ϵ).x(t),S(t)RN are the fast and slow state vectors, respectively; the vector fields f1C1(RN) and f2C(RN) are known and satisfy f1(0)=f2(0)=0. The singular perturbation parameter ϵ is

Competitive neural network with long and short term memory dynamics

The general neural network equations describing the temporal evolution of the unperturbed short term memory (STM) and long term memory (LTM) states for the jth neuron of an N-neuron network are ṁij=mij+yif(xj)ϵẋj=ajxj+i=1NDijf(xi)+Bjk=1pmkjyk where xj (j=1,,N) is the current activity level, aj is the time constant of the neuron, Bj is the contribution of the external stimulus term, f(xi) (i=1,,N) is the neuron’s output, yk (k=1,,p) is the external stimulus and mij is the synaptic

Conclusions

The dynamical behavior of biological networks, subject to fluctuations based on the theory of uncertain singularly perturbed systems, have been analyzed. The nominal system was nonlinear and it was assumed that the fluctuations are bounded. In particular, the results obtained have been used to study the robustness properties of a certain class of two-time-scale biological networks. In this sense the robust stability results for the perturbed network model have been presented and the conditions

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An abbreviated version of some portions of this article appeared as part of the IJCNN 2009 Conference Proceedings, published under IEEE copyright.

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