1 Introduction
Recurrently connected neural networks, also known as the Hopfield neural networks, have been extensively studied in past decades and found many applications in different areas. Such applications heavily depend on the dynamical behaviors of the system. Therefore, analysis of the dynamics is a necessary step for practical design of neural networks.
The dynamical behaviors of continuoustime recurrently asymmetrically connected neural networks (CTRACNN) have been studied at the very early stage of neural network research. For example, multistable and oscillatory behaviors were studied by Amari (1971, 1972) and Wilson & Cowan (1972). Chaotic behaviors were studied by Sompolinsky, & Crisanti (1988). Hopfield, & Tank (1984, 1986) studied stability of symmetrically connected networks and showed their practical applicability to optimization problems. It should be noted that Cohen and Grossberg, see Cohen, & Grossberg (1983) gave more rigorous results on the global stability of networks.
The global stability of symmetrically connected networks described by differential equations has now been well established. See Chen (1999); Chen, & Amari (2001); Chen, & Lu (2002); Fang, & Kincaid (1996); Forti, & Marini (1994); Hirsch (1989); Kaszkurewicz, & Bhaya (1994); Kelly (1990); Li, Michel, & Porod (1988); Matsuoka (1992); Yang, & Dillon (1994) and the references therein. More related to the present paper, the previous paper (Liu, Lu, & Chen, 2011) addressed the global selfsynchronization of general continuoustime asymmetrically connected recurrent networks and discussed the independent identically distributed switching process on the selecting the timevarying parameters in detail.
However, in applications, discrete iteration is popular to be employed to realize neural network process, rather than continuoustime equations. Generally, synchronization analysis for differential equations cannot be applicable to the discretetime situation. There are several papers (Jin, Nikiforuk, & Gupta, 1994; Jin, & Gupta, 1999; Wang, 1997) that discussed different types of discretetime neural networks, where the step sizes were constants. However, in Liu, Chen, & Yuan (2012); Manuel, & Tabuada (2011); Seyboth, Dimarogonas, & Johansson (2013); Wang, & Lemmon (2008), these papers pointed out that the constant timestep size was costly. This motivates us to design adaptive step sizes for synchronization of asymmetric recurrent timevarying neural network.
Moreover, the discretization is related to the concept of sampleddata control. There are a number of papers discussing dynamics of neural networks, using sampleddata control. The papers (Lam, & Leung, 2006; Wu, Shi & Su, 1972; Zhu, & Wnag, 2011) applied the sampleddata control technique towards stabilization of threelayer fully connected feedforward neural networks. In Chandrasekar, Rakkiyappan, Rihan, & Lakshmanan (2014); Jung, Park, & Lee (2014); Lee, Park, Kwon, & Lee (2013); Liu, Yu, Cao, & Chen (2015); Rakkiyappan, Chandrasekar, Park, & Kwon (2014), the authors used sampleddata control strategy for exponential synchronization for the neural networks with Markovian jumping parameters and time varying delays. Rakkiyappan, Sakthivel, Park, & Kwon (2013)
discussed state estimation for Markovian jumping fuzzy cellular neural networks with probabilistic timevarying delays with sampleddata.
The purpose of this paper is to give a comprehensive analysis on outsynchronization of the discretetime recurrently asymmetrically connected timevarying neural networks. We propose two schemes of discretizations, named centralized and decentralized discretization respectively, and present sufficient conditions for the global outsynchronization. The common step size for every neuron in centralized discretization but in decentralized discretization process, the distributed step size for each neuron is used to guarantee that any two trajectories from different initial values converge together.
2 Preliminaries and problem formulation
In this section, we provide the models of asymmetric recurrent neural networks with datasampling, and some notations. The continuoustime version of the recurrent connected neural networks is described by the following differential equations
(1) 
where , and are piecewise continuous and bounded, , and satisfies
(2) 
for all , where is a constant and .
In the centralized datasampling strategy, the continuoustime system (1) is rewritten as
(3) 
for . The increasing time sequence ordered as is uniform for all the neuron . Each neuron broadcasts its state to its outneighbours and receives its inneighbours’ states information at time .
Comparatively, in the decentralized datasampling strategy, Eq. (1) is rewritten as the following pushbased decentralized system
(4) 
for . The increasing time sequence order as is distributed for the neuron . Every neuron pushes its state information to its outneighbours at time when it updates its state. It receives its inneighbours’ state information at time when its neighbour neuron renews it state.
To begin the discussion, we give the following three norms of and recall the definition of outsynchronization proposed in Wu, Zheng, & Zhou (2009).
Definition 1
Let be a positive constant and we can define three generalized norms as follow

norm:

norm:

norm:
where is a vector.
Definition 2
Consider any two trajectories and starting from different initial values and of the following system
(5) 
The system (5) is said to achieve outsynchronization if there exists a controller for the two trajectories and such that
Other major notations which will be used throughout this paper are summarized in the following definition.
Definition 3
Let be a positive constant and then we define
and
where and .
Because of the boundedness of the functions, it can be seen that and are bounded for all . That is, there exist positive constants and such that
with .
3 Structuredependent datasampling principle
In this section, we provide several the structurebased datasampling rules for the next triggering time point at which the neurons renew their states and the control signals.
3.1 Structuredependent centralized datasampling
For any neuron , consider two trajectories and of the system (3) starting from different initial values. Denote with . And it holds
(6) 
where for all , and .
The following theorem gives conditions that guarantee the system (3) reaches outsynchronization via  norm.
Theorem 1
Let and be constants with and . Suppose that there exist , such that for all and . Set an increasing timepoint sequence as
(7) 
. Then the system (3) reaches outsynchronization.
Proof. From the condition , one can see that
which implies that exists for all and . Thus, one can further see
(8) 
and
(9) 
for all and . Furthermore, we have
Consider for each , and we have
with
which implies for all and , according to (2). Note
From (8), one can see
which leads
(10) 
Then, it follows
(11) 
The last equality holds due to (10). Thus, according to the rule (7) and (9), which implies
since the equality in (7) occurs at , thus we have
which implies
In addition, for each , from the rule (7) and the condition , inequality (11) implies that for each , . Hence, it holds
The outsynchronization of system (3) is proved.
The proofs of the following results are analog to Theorem 3 but via and norm. Their proofs are similar to that of Theorem 3, which can be found in Zheng, Chen, & Lu (2015) and so neglected in the present paper.
Proposition 1
Let and be constants with and . Suppose that there exist , such that for all and . Set an increasing timepoint sequence as
(12) 
. Then the system (3) reaches outsynchronization.
Proposition 2
Let and be constants with and . Suppose that there exist , such that for all and . Set an increasing timepoint sequence as
(13) 
. Then the system (3) reaches outsynchronization.
To explain the independence of the results via three norms, we give out the following example. Denote
Let with
when and
when .
In the first time interval , we have found that when
it holds
where . In the second time interval , we can find that when
it follows
However, to maintain , we have to solve the following inequalities
that is
One can see that there is no such solution of and .
3.2 Structuredependent pushbased decentralized datasampling
For each neuron , consider two trajectories and of system (4) starting from different initial values. Denote with . It follows
(14) 
where for all , and .
The following theorem and propositions give conditions that guarantee the convergence of system (14) via three generalized norms (, and ).
Theorem 2
Let and be constants with and . Suppose that there exist such that for all and . Set as the triggering time points as
(15) 
for and . Then the system (4) reaches outsynchronization.
Proof. For each , let . Similar to the arguments up to (11) in the proof of Theorem 1, one can derive the following inequality immediately:
(16) 
From the arguments of (9), one can conclude
(17) 
in an analog way.
Let be an increasing sequence such that and , which implies that for each neuron , equality in the rule (2) occurs at least once. Thus, we have
Consider for any neuron at triggering time where , and we have
By the inequality (2), it holds
Based on the triggering rule (2), we can obtain
which means
For any time , the state becomes
where . Thus,
for any and , which implies
The proof for the outsynchronization of system (4) is completed.
Proposition 3
Let be a constant and . Set as the time points such that
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