Periodicity and exponential stability of discrete-time neural networks with variable coefficients and delays
© Xu and Wu; licensee Springer 2013
Received: 19 March 2013
Accepted: 10 July 2013
Published: 25 July 2013
Discrete analogues of continuous-time neural models are of great importance in numerical simulations and practical implementations. In the current paper, a discrete model of continuous-time 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.
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].
Time-delays 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 , 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, time-varying and distributed delays, respectively.
Meanwhile, in numerical simulations and practical implementations, discretization of continuous-time 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 discrete-time 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  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 continuous-time 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 , inequality technique and matrix analysis  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, discrete-time 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
where , , ; , is the state of the i th neuron at time t; the continuous function represents the neuron charging time, is the strength of the j th unit on the i unit at time ; denotes the activation function of the neuron, which satisfies the global Lipschitz condition; denotes the transmission delay along the axon of the j th unit; the continuous function is the external input on the i th neuron at time t; the initial value function is bounded and continuous on . Research into dynamics such as stability and periodic oscillations for this model has been extensively carried out . Now we will focus on the dynamics of its discrete analogue.
Set Z to be the set of integers and the set of nonnegative integers; let represent the set of integers between a and b with , , namely, .
where is bounded on .
Note that and for small . It could be showed that the discrete-time analogue (3) converges to the continuous-time model (1) as .
To investigate the stability and periodic oscillations of system (3), we make further assumptions.
(H1) Suppose that , , and are all ω-periodic functions; moreover, , , with ω and k being positive integers, for .
for any ; further, , .
where , .
For any , a solution of system (3) and (4) is a vector-valued function satisfying system (3) and initial conditions (4) for . In this paper, it is always assumed that the neural model (3) and (4) admits a solution represented by or simply .
To analyze the existence and stability of periodic solutions for system (3), M-matrix theory is employed. Some notations and terminologies are given below. For more details, please refer to .
Definition 2.1 
Matrix is said to be a nonsingular M-matrix if (i) , (ii) for , (iii) , .
Lemma 2.1 
Lemma 2.2 
- (i)There exists a constant vector , with , , such that
The real parts of all eigenvalues of Λ are positive.
- (iii)There exists a symmetric positive definite matrix W such that
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 ω-periodic solutions are established. Let us first introduce the continuation theorem due to Gaines and Mawhin .
Let X and Y be two real Banach spaces, let be a Fredholm operator of index zero, and let , be continuous projectors such that , , and , . Denote by the restriction of L on ; by the inverse of ; by the algebraic and topological isomorphism of ImQ onto KerL, due to the same dimensions of these two subspaces.
Lemma 3.1 (Continuation theorem )
for each , , ;
for each , ;
Then admits at least one solution in .
with which X, Y are Banach spaces, here is the Euclidean norm.
where I is the identity matrix. From periodicity and positiveness, it holds that , .
Lemma 3.2 Under hypothesis (H3), matrix Γ is a nonsingular M-matrix.
it follows from (H3) and Lemma 2.2 that Γ is a nonsingular M-matrix. □
then system (3) admits at least one ω-periodic solution.
hence ImL is closed in X, and , that is, L is a Fredholm operator of index zero.
In this way, , , , and the isomorphism J from ImQ onto KerL is taken to be the identity map.
Consequently, by the Lebesgue convergence theorem, QN and are continuous, and via the Arzela-Ascoli theorem, and are compact for any open bounded set , namely, N is L-compact on .
where , .
that is, condition (b) in Lemma 3.1 holds.
Hence condition (c) in Lemma 3.1 is verified. By Lemma 3.1, we conclude that system (3) admits at least one ω-periodic solution. This completes the proof. □
4 Exponential stability of periodic solutions
Theorem 4.1 Suppose that all conditions in Theorem 3.1 hold except that (H3) is replaced by
for all .
The proof is complete. □
for all .
That implies that the ω-periodic solution of system (3) and (4) is globally exponentially stable. The proof is completed. □
Remark Frequently activation functions 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  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.
Then system (3) admits a unique ω-periodic solution, which is globally exponentially stable, that is, all other solutions converge to it exponentially as .
Corollary 4.2 Under conditions (H1), (H2) and , if the matrix is a nonsingular M-matrix, where , , then system (3) admits a unique ω-periodic solution, which is globally exponentially stable.
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.
and, further, the matrix is a nonsingular M-matrix, 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 discrete-time neural models have been given. For example, Chen et al.  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 Max-cut problems and cellular channel assignments; and Yashtini et al.  gave the discrete model for nonlinear convex programming. Moreover, the discrete-time cellular neural networks for associate memories with learning and forgetting capabilities [36, 37] also were established. So, discrete neural networks have extensive uses in real-life applications.
The example is a discrete network of two neurons with self-connection, 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 larger-scale networks.
It is the Hopfield network with associate memory and data storage capability . 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.
In the current paper, a class of discrete-time 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 discrete-time analogue model well preserves the dynamical behaviors from the continuous version.
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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