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Periodicity and exponential stability of discretetime neural networks with variable coefficients and delays
Advances in Difference Equations volume 2013, Article number: 226 (2013)
Abstract
Discrete analogues of continuoustime neural models are of great importance in numerical simulations and practical implementations. In the current paper, a discrete model of continuoustime neural networks with variable coefficients and multiple delays is investigated. By Lyapunov functional, continuation theorem of topological degree, inequality technique and matrix analysis, sufficient conditions guaranteeing the existence and globally exponential convergence of periodic solutions are obtained, without assuming the boundedness and differentiability of activation functions. To show the effectiveness of our method, an illustrative example is presented along with numerical simulations.
MSC:34D23, 34K20, 39A12, 92B20.
1 Introduction
Various neural networks have been so far proposed, and they attracted extensive interest of researchers from various fields, since they play important roles and have found successful applications in the fields such as pattern recognition, signal and image processing, nonlinear optimization problems, parallel computation, and other engineering areas; see, for example, [1–8]. The dynamical behaviors in neural networks, such as the existence and their asymptotic stability of equilibria, periodic solutions, bifurcations and chaos, have been the most active areas of research and have been extensively studied over the past years [9–21].
Timedelays in interactions between neurons are frequently unavoidable due to the finite transmission speed of signals among neurons, and they cause instability, divergence and oscillations in neural networks [22], so it is necessary to introduce time delays into the neural models. Numerous sufficient conditions ensuring the stability have been given for neural models with discrete, timevarying and distributed delays, respectively.
Meanwhile, in numerical simulations and practical implementations, discretization of continuoustime models is necessary and of great importance. Certainly, to faithfully reflect the dynamical behaviors of continuous systems, the discrete analogues should inherit the dynamical characteristics of continuous counterparts [13, 23]. The ways to derive the discretetime analogues from continuous versions are diverse, but most of them cannot keep the original dynamics and display more complicated behaviors. To this end, an implicit scheme has been put forward [13] to derive the discrete analogues. For discrete models under this scheme, such as discrete Hopfield, bidirectional associate memory and cellular neural networks, several authors [24–29] have studied the existence and exponential stability of equilibria and periodic solutions. However, they are mainly concerned with the models of constant coefficients. Research on the discrete models with variable coefficients and delays is very rare. Since the neuron charging time, interconnection weights and external inputs often change during the course of time, neural models with temporal structure of neural activities are much closer to real systems, and hence studies on such systems and their discrete versions are of great practical and theoretical value.
Motivated by the above discussions, we present a discrete analogue of continuoustime neural networks with variable coefficients and delays. Here the activation functions are not assumed to be bounded, as opposed to those in [25–28]. To deal with this general model, a suitable and effective Lyapunov functional is constructed. Continuation theorem of topological degree [30], inequality technique and matrix analysis [31] are employed to obtain the sufficient conditions guaranteeing the existence and globally exponential stability of periodic solutions. As we see, these sufficient conditions are less conservative and easy to verify. Further, no restrictions of the differentiability and monotonicity are imposed on activation functions. Also, note that the discrete models admit the common dynamical behaviors with continuous versions. That implies they preserve the dynamics very well from continuous versions. To show the effectiveness of our results, an illustrative example along with numerical simulations is presented.
The paper is organized as follows. In Section 2, discretetime neural networks with variable coefficients and delays are formulated. Some assumptions and mathematical preliminaries are also given. Section 3 is devoted to the existence of periodic solutions. In Section 4, the exponential convergence of this discrete model is discussed. To show the effectiveness of the method, an illustrative example is presented in Section 5. Some conclusions are drawn in Section 6.
2 Mathematical preliminaries
Continuoustime neural networks with timevarying coefficients and delays read as follows:
with initial values
where $i=1,\dots ,m$, $t\in {R}^{+}=[0,+\mathrm{\infty})$, $0\le \sigma (t)\le \tau $; $x=col({x}_{1},\dots ,{x}_{m})\in {R}^{m}$, ${x}_{i}(t)$ is the state of the i th neuron at time t; the continuous function ${a}_{i}(t)$ represents the neuron charging time, ${b}_{ij}(t)$ is the strength of the j th unit on the i unit at time $t\sigma (t)$; ${f}_{j}$ denotes the activation function of the neuron, which satisfies the global Lipschitz condition; $\sigma (t)$ denotes the transmission delay along the axon of the j th unit; the continuous function ${I}_{i}(t)$ is the external input on the i th neuron at time t; the initial value function ${\varphi}_{i}(s)$ is bounded and continuous on $[\tau ,0]$. Research into dynamics such as stability and periodic oscillations for this model has been extensively carried out [18]. Now we will focus on the dynamics of its discrete analogue.
Set Z to be the set of integers and ${Z}^{+}$ the set of nonnegative integers; let $\mathbf{N}(a,b)$ represent the set of integers between a and b with $a\le b$, $a,b\in Z$, namely, $\mathbf{N}(a,b)=\{a,a+1,\dots ,b\}$.
Reformulate the continuoustime model (1) by equations with piecewise constant arguments of the form
where $i=1,\dots ,m$, $t\in ([t/h]h,[t/h]h+h)$, $h>0$ is the discretization step size and $[t/h]$ is the integer part of $t/h$. Let
where $n=0,1,2,\dots $ , then one has
Integrate over the interval $[nh,t)$ to get
let $t\to (n+1)h$, then
where
The discrete model (3) is endowed with initial values
where ${\varphi}_{i}(l)$ is bounded on $\mathbf{N}(k,0)$.
Note that ${\theta}_{i}(h)>0$ and ${\theta}_{i}(h)\approx h+O({h}^{2})$ for small $h>0$. It could be showed that the discretetime analogue (3) converges to the continuoustime model (1) as $h\to 0$.
To investigate the stability and periodic oscillations of system (3), we make further assumptions.
(H1) Suppose that ${a}_{i}(n)$, ${b}_{ij}(n)$, ${I}_{i}(n)$ and $k(n)$ are all ωperiodic functions; moreover, ${a}_{i}(n)>0$, $0\le k(n)\le k$, with ω and k being positive integers, for $i=1,\dots ,m$.
(H2) Assume that a function ${f}_{j}$ satisfies the Lipschitz condition, i.e., there exists a constant ${F}_{j}>0$ such that
for any $\xi ,\eta \in R$; further, ${f}_{j}(0)=0$, $j=1,\dots ,m$.
(H3) Suppose that there exist constants ${\lambda}_{i}>0$ such that
where ${a}_{i}^{}={min}_{0\le n\le \omega 1}{a}_{i}(n)$, $i=1,\dots ,m$.
For any $\varphi =({\varphi}_{1},\dots ,{\varphi}_{m})$, a solution of system (3) and (4) is a vectorvalued function $x:{Z}^{+}\to {R}^{m}$ satisfying system (3) and initial conditions (4) for $n\in {Z}^{+}$. In this paper, it is always assumed that the neural model (3) and (4) admits a solution represented by $x(n,\varphi )$ or simply $x(n)$.
For later convenience, throughout this paper $\overline{f}$ represents the mean value of a function $f(n)$ over $[0,\omega 1]$. Denote by ${g}^{+}$, ${g}^{}$ the maximum and the minimum of a function $g(n)$ over $[0,\omega 1]$, respectively, i.e.,
To analyze the existence and stability of periodic solutions for system (3), Mmatrix theory is employed. Some notations and terminologies are given below. For more details, please refer to [31].
Definition 2.1 [31]
Matrix $\mathrm{\Lambda}=({a}_{ij})$ is said to be a nonsingular Mmatrix if (i) ${a}_{ii}>0$, (ii) ${a}_{ij}\le 0$ for $i\ne j$, (iii) ${\mathrm{\Lambda}}^{1}\ge 0$, $i,j=1,\dots ,m$.
Lemma 2.1 [31]
If Λ is a nonsingular Mmatrix and $\mathrm{\Lambda}y\le h$, then
Lemma 2.2 [31]
Let Λ be an $m\times m$ matrix with nonpositive offdiagonal elements. Then Λ is a nonsingular Mmatrix if and only if one of following statements holds true:

(i)
There exists a constant vector $\xi ={({\xi}_{1},\dots ,{\xi}_{m})}^{T}$, with ${\xi}_{i}>0$, $i=1,\dots ,m$, such that
$$\mathrm{\Lambda}\xi >0.$$ 
(ii)
The real parts of all eigenvalues of Λ are positive.

(iii)
There exists a symmetric positive definite matrix W such that
$$\mathrm{\Lambda}W+W{\mathrm{\Lambda}}^{T}$$
is positive definite.

(iv)
All of the principal minors of Λ are positive.
3 Periodic solutions
Next we investigate the existence and global exponential stability of periodic solutions of system (3). By the continuation theorem of topological degree, the Lyapunov functional and analytic techniques such as matrix analysis, the existence and its global exponential stability of ωperiodic solutions are established. Let us first introduce the continuation theorem due to Gaines and Mawhin [30].
Let X and Y be two real Banach spaces, let $L:DomL\cap X\to Y$ be a Fredholm operator of index zero, and let $P:X\to X$, $Q:Y\to Y$ be continuous projectors such that $ImP=KerL$, $KerQ=ImL$, and $X=KerL\oplus KerP$, $Y=ImL\oplus ImQ$. Denote by ${L}_{P}$ the restriction of L on $KerP\cap DomL$; by ${K}_{P}:ImL\to KerP\cap DomL$ the inverse of ${L}_{P}$; by $J:ImQ\to KerL$ the algebraic and topological isomorphism of ImQ onto KerL, due to the same dimensions of these two subspaces.
Lemma 3.1 (Continuation theorem [30])
Let $\mathrm{\Omega}\subset X$ be an open bounded set and let $N:X\to Y$ be a continuous operator which is Lcompact on $\overline{\mathrm{\Omega}}$. Suppose that

(a)
for each $\lambda \in (0,1)$, $x\in \partial \mathrm{\Omega}\cap DomL$, $Lx\ne \lambda Nx$;

(b)
for each $x\in \partial \mathrm{\Omega}\cap KerL$, $QNx\ne 0$;

(c)
$deg\{JQN,\mathrm{\Omega}\cap KerL,0\}\ne 0$.
Then $Lx=Nx$ admits at least one solution in $\overline{\mathrm{\Omega}}\cap DomL$.
To achieve our goal, take
endowed with the norm
with which X, Y are Banach spaces, here $\cdot $ is the Euclidean norm.
Set
where I is the identity matrix. From periodicity and positiveness, it holds that ${a}_{i}^{+}\ge {a}_{i}(n)\ge {a}_{i}^{}>0$, $i=1,\dots ,m$.
Lemma 3.2 Under hypothesis (H3), matrix Γ is a nonsingular Mmatrix.
Proof Since
it follows from (H3) and Lemma 2.2 that Γ is a nonsingular Mmatrix. □
Theorem 3.1 Suppose that hypotheses (H1)(H3) hold. If
then system (3) admits at least one ωperiodic solution.
Proof Let $L:DomL\cap X\to X$ be defined by
and let the i th component of the mapping $N:X\to X$ be defined, respectively, by
With these notations, system (3) is rewritten into the form
It is not difficult to see that
hence ImL is closed in X, and $dimKerL=codimImL=m$, that is, L is a Fredholm operator of index zero.
Define the linear continuous projectors $P,Q:X\to X$ by
In this way, $ImP=KerL$, $ImL=KerQ$, $X=KerL\oplus KerP=ImL\oplus ImQ$, and the isomorphism J from ImQ onto KerL is taken to be the identity map.
Clearly, the mapping ${L}_{P}:DomL\cap KerP\to ImL$ is onetoone and onto, so invertible. Its inverse ${K}_{P}:ImL\to DomL\cap KerP$ is defined as
Consequently, by the Lebesgue convergence theorem, QN and ${K}_{P}(IQ)N$ are continuous, and via the ArzelaAscoli theorem, $QN(\overline{\mathrm{\Omega}})$ and ${K}_{P}(IQ)N(\overline{\mathrm{\Omega}})$ are compact for any open bounded set $\mathrm{\Omega}\subset X$, namely, N is Lcompact on $\overline{\mathrm{\Omega}}$.
For a certain $\lambda \in (0,1)$, suppose that $x\in DomL\cap X$ is a solution of
equivalently,
Now one could get the estimates as follows:
Therefore, we have
for $i=1,\dots ,m$. Set
then inequality (6) is equivalent to
From Lemma 2.2, we know that Γ is a nonsingular Mmatrix, then from Lemma 2.1, it holds that
That means ${max}_{0\le n\le \omega 1}{x}_{i}(n)$ is bounded, i.e., there exist constants ${\alpha}_{i}$ such that
Take $\beta >0$ such that
where $M={\sum}_{i=1}^{m}{\lambda}_{i}^{1}{\alpha}_{i}+\beta $, $\lambda ={min}_{1\le i\le m}\{{\lambda}_{i}\}$.
Set
according to the above discussions, $Lx\ne \lambda Nx$ for $x\in \partial \mathrm{\Omega}\cap Dom\; L$, $\lambda \in (0,1)$. So, condition (a) in Lemma 3.1 is satisfied. When $x\in \partial \mathrm{\Omega}\cap KerL$, x is a constant vector in ${R}^{m}$, with $\parallel x\parallel ={\sum}_{i=1}^{m}{\lambda}_{i}^{1}{x}_{i}=M$, then $QNx$ is expressed as $QNx={({(QNx)}_{1}(n),\dots ,{(QNx)}_{m}(n))}^{T}$, with
where $i=1,\dots ,m$. Since
and further
in this way,
Consequently,
that is, condition (b) in Lemma 3.1 holds.
Let $\mathrm{\Phi}:KerL\times [0,1]\to X$ be defined by
where $\varphi (x)=({\overline{a}}_{1}{\theta}_{1}(h){x}_{1},\dots ,{\overline{a}}_{m}{\theta}_{m}(h){x}_{m})$. When $x\in \partial \mathrm{\Omega}\cap KerL$, it follows that
That means
As a result, the homotopy invariance implies
Hence condition (c) in Lemma 3.1 is verified. By Lemma 3.1, we conclude that system (3) admits at least one ωperiodic solution. This completes the proof. □
4 Exponential stability of periodic solutions
By Theorem 3.1, system (3) has at least an ωperiodic solution ${x}^{\ast}(n)={({x}_{1}^{\ast}(n),\dots ,{x}_{m}^{\ast}(n))}^{T}$. Clearly, if ${x}^{\ast}(n)$ is exponentially stable, then the ωperiodic solution of system (3) is unique. Now we will investigate the exponential stability of periodic solutions. Set $u(n)=x(n){x}^{\ast}(n)$, then system (3) is equivalent to
Theorem 4.1 Suppose that all conditions in Theorem 3.1 hold except that (H3) is replaced by
(H4) Suppose that there exist constants ${\lambda}_{i}>0$ such that
then the ωperiodic solution of system (3) is globally exponentially stable in the sense that there exist constants $\eta >1$ and $C>1$ such that for any solution $x(n,\varphi )$ of system (3) and (4) with initial condition ϕ, it holds that
for all $n\in {Z}^{+}$.
Proof Define the function ${G}_{i}:R\to R$ by
for $i=1,\dots ,m$. From (H4), we have
From the continuity of functions ${G}_{i}$, there must be a number $\eta >1$ such that
that is,
for $i=1,\dots ,m$. From (9) and (H2), one has
Set ${U}_{i}(n)={\eta}^{n}\frac{{u}_{i}(n)}{{\theta}_{i}(h)}$, then it holds from (11) that
where $n\in {Z}^{+}$. Define a Lyapunov functional $V(n)=V({U}_{1},\dots ,{U}_{m})(n)$ as follows:
where
To investigate the exponential stability of an ωperiodic solution ${x}^{\ast}(n)$ by the Lyapunov functional $V(n)$, it is necessary to calculate the difference $\mathrm{\Delta}V(n)=V(n+1)V(n)$ along the solutions of (12). From (12), we have
Since
one obtains
Therefore, we have
In view of (10), it follows that $\mathrm{\Delta}V(n)\le 0$ for all $n\in {Z}^{+}$, which means that $V(n)\le V(0)$ for $n=1,2,3,\dots $ . Note that
so we have
where $n\in {Z}^{+}$, $\lambda ={min}_{1\le i\le m}\{{\lambda}_{i}\}$ and
The proof is complete. □
Theorem 4.2 Suppose that all conditions in Theorem 3.1 hold, then the ωperiodic solution of system (3) is globally exponentially stable in the sense that there exist constants $\eta >1$ and ${C}^{\ast}>0$ such that for any solution $x(n,\varphi )$ of system (3) and (4) with initial conditions ϕ, it holds that
for all $n\in {Z}^{+}$.
Proof Define the function ${G}_{i}^{\ast}:R\to R$ by
From (H3), we have
From the continuity of functions ${G}_{i}^{\ast}$, there must be a number $\eta >1$ such that
that is,
From (9) and (H2), one has
Set ${V}_{i}(n)={\eta}^{n}{\lambda}_{i}^{1}{u}_{i}(n)$, then it holds that
where $n\in {Z}^{+}$. Let $M={max}_{1\le i\le m}\{{sup}_{l\in \mathbf{N}(k,0)}{\lambda}_{i}^{1}{u}_{i}(l)\}$. It is clear that ${V}_{i}(l)\le M$ for $l\in \mathbf{N}(k,0)$, $i=1,\dots ,m$. We claim that
Otherwise, there should be an index r and a positive integer ${n}_{1}$ such that
and
That is, ${n}_{1}$ is the first time that inequality (16) is violated. Meanwhile, by (15) and (13), one has
That leads to a contradiction. Therefore, the assertion (16) is true. Consequently,
That implies that the ωperiodic solution ${x}^{\ast}(n)$ of system (3) and (4) is globally exponentially stable. The proof is completed. □
Remark Frequently activation functions ${f}_{j}$ are assumed to be bounded and monotonic; see, for instance, [18, 28]. However, no restrictions of boundedness, monotonicity and differentiability are imposed in this paper. Moreover, the conditions ensuring exponential stability are less conservative and easy to verify. Also, it could be noted [18] that continuous model (1) admits the common behaviors with system (3).
From Theorems 3.1, 4.1, 4.2 and Lemma 2.2, some corollaries could be immediately derived.
Corollary 4.1 Assume that hypotheses (H1) and (H2) hold and $\kappa >0$; further assume that one of the following conditions holds:

(i)
${a}_{i}^{}(1+{k}^{+}{k}^{}){F}_{i}{\sum}_{j=1}^{m}{b}_{ji}^{+}>0$;

(ii)
${a}_{i}^{}{\sum}_{j=1}^{m}{b}_{ij}^{+}{F}_{j}>0$, $i=1,\dots ,m$.
Then system (3) admits a unique ωperiodic solution, which is globally exponentially stable, that is, all other solutions converge to it exponentially as $n\to \mathrm{\infty}$.
Corollary 4.2 Under conditions (H1), (H2) and $\kappa >0$, if the matrix $ADF$ is a nonsingular Mmatrix, where $A=diag\{{a}_{1}^{},\dots ,{a}_{m}^{}\}$, $D={({b}_{ij}^{+})}_{m\times m}$, then system (3) admits a unique ωperiodic solution, which is globally exponentially stable.
When ${a}_{1}(n)\equiv {a}_{i}$, ${b}_{ij}(n)\equiv {b}_{ij}$, ${I}_{i}(n)\equiv {I}_{i}$, $k(n)\equiv k$, then the matrices $A=diag\{{a}_{1},\dots ,{a}_{m}\}$, $D={({b}_{ij})}_{m\times m}$ and system (3) reduces to the model with constant coefficients and delays, that is,
Since the equilibrium could be viewed as the periodic solution of arbitrary period, as a consequence of Theorems 3.1, 4.1 and 4.2, we obtain the following corollary.
Corollary 4.3 If ${f}_{j}$ satisfies hypothesis (H2), ${a}_{i}>0$, $k>0$,
and, further, the matrix $ADF$ is a nonsingular Mmatrix, then system (17) has a unique equilibrium, which is globally exponentially stable.
5 Numerical results
Note that when neural networks are applied to practical problems such as image processing, pattern recognition, artificial intelligence, computer simulations, and so on, discrete versions of models are needed, since the information is discrete in nature and processing procedures occur in discrete steps.
Now some algorithms based on discretetime neural models have been given. For example, Chen et al. [32] put forward an image processing method based on discrete neural models, which were implemented on circuits; Wang et al. [33, 34] proposed the discrete Hopfield neural models for Maxcut problems and cellular channel assignments; and Yashtini et al. [35] gave the discrete model for nonlinear convex programming. Moreover, the discretetime cellular neural networks for associate memories with learning and forgetting capabilities [36, 37] also were established. So, discrete neural networks have extensive uses in reallife applications.
To show the effectiveness of the obtained theoretical results, an illustrative example is given. Consider the following discretetime neural network:
The example is a discrete network of two neurons with selfconnection, which is of Hopfield type. The model is often implemented in practical applications such as image processing, pattern recognition and artificial intelligence. Such networks are the prototypes to understand the dynamics of largerscale networks.
The corresponding continuous model is
It is the Hopfield network with associate memory and data storage capability [38]. To measure the information transmitted among neurons, i.e., inputs and outputs, discrete samplings are necessary on discrete time instances. Frequently, periodic samplings are adopted. So, under the proposed discretization scheme, the discrete version of neural model follows.
Now take
Activation functions ${f}_{1}$, ${f}_{2}$ are chosen to be hyperbolic tangent and inverse tangent ones, respectively, which are of sigmoid symmetric type. They are frequently used in neural networks and work very well in applications. The coefficients ${a}_{i}$, ${b}_{ij}$ ($i,j=1,2$) are chosen to be periodic functions. Here we take trigonometric functions. It is not difficult to see that (H1) and (H2) are satisfied by this discretetime neural network. The parameters are as follows:
and
When ${\lambda}_{1}$, ${\lambda}_{2}$ are set to be 1, $\frac{1}{2}$, respectively, (H3) is also true, so from Theorems 3.1 and 4.2, it admits a unique periodic solution, with all other solutions converging to it exponentially as $n\to \mathrm{\infty}$ (see Figures 13).
From Figures 13, note that all the solutions tend to the unique periodic solution. So, the unique periodic solution exists and it is exponentially stable. The trajectories of a continuous model are showed in Figure 4. Note that it admits a unique periodic solution. Also note that the discrete analogue model well preserves the dynamics of the corresponding continuous one.
6 Conclusions
In the current paper, a class of discretetime neural networks has been studied. Using the coincidence degree, the Lyapunov functional and matrix analysis, the existence and its global exponential stability of a periodic solution have been established for this model, assuming no boundedness, monotonicity and differentiability of activation functions. The obtained results are less conservative and will be of practical use for applying neural models. Also note that the discretetime analogue model well preserves the dynamical behaviors from the continuous version.
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Acknowledgements
This study is supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20093401120001), the Natural Science Foundation of Anhui Province (No. 11040606M12) and the Natural Science Foundation of Anhui Education Bureau (No. KJ2010A035), the 211 project of Anhui University (No. KJJQ1102).
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RW directed the study, helped inspection, established the models, carried out the results of this article and drafted the manuscript. HX performed the numerical simulation. All the authors read and approved the final manuscript.
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Xu, H., Wu, R. Periodicity and exponential stability of discretetime neural networks with variable coefficients and delays. Adv Differ Equ 2013, 226 (2013). https://doi.org/10.1186/168718472013226
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Keywords
 coincidence degree
 discrete neural networks
 variable coefficient
 exponential stability
 periodic solution
 Mmatrix