Pinning synchronization of memristor-based neural networks with time-varying delays
Introduction
The theoretical prototype of memristor which is regarded as the fourth circuit element (the other three are resistor, inductor and capacitor), was proposed by Professor Chua in 1971 (Chua, 1971). However, few researchers paid attention to it because it was thought as an ideal element and may only exist in the literature. This situation did not change until the practical memristor was realized with nanometer dimensions by Hewlett-Packard Lab in 2008 (Strukov, Snider, Stewart, & Williams, 2008). Then, more and more researchers from different branches start to focus on this field. The results about memristor mainly focus on the memristor devices Corinto et al. (2011), Itoh & Chua (2008), Kim et al. (2012) and memristor systems Adhikari et al. (2012), Itoh & Chua (2009), Pershin & Di Ventra (2010). The memristor-based neural networks (MNNs) are constructed by memristors in which the connection between two neurons is implemented by a memristor to replace the conventional resistor in Liu et al. (2008), Sanchez & Perez (2003). The MNNs are state-dependent nonlinear systems whose weights are switchable and depend on the states of the systems.
Synchronization is a typical collective behavior in nature and synchronization of neural network is an attractive topic for researchers due to its applications in secure communication Yang & Cao (2007), Zhang et al. (2006), Zhang et al. (2007), information science Cheng et al. (2006), Hu et al. (2017), Li et al. (2014), Wang et al. (2010), Zhang et al. (2010) and image processing (Prakash, Balasubramaniam, & Lakshmanan, 2016). Similarly, the synchronization of memristor-based systems can also be applied to secure communications due to its nonlinear and chaotic characteristics Lin et al. (2015), Sun et al. (2013). Up to now, there are already some results about synchronization of MNNs Abdurahman et al. (2016), Bao et al. (2015), Chandrasekar & Rakkiyappan (2016), Han et al. (2016), Li & Wei (2016), Liu et al. (2016a), Liu et al. (2016b), Mathiyalagan et al. (2015), Wang et al. (2016), Wang et al. (2015), Wen et al. (2013), Wen et al. (2015), Wen et al. (2014a), Wen et al. (2014b), Wu et al. (2015), Wu et al. (2012), Yang et al. (2016a), Yang et al. (2016b), Zhang et al. (2016), Zhang & Shen (2014), Zhang et al. (2013). In Wu et al. (2012), Wu et al. investigated a class of memristor-based recurrent neural networks and proposed some sufficient conditions to guarantee the exponential synchronization based on differential inclusions theory and Lyapunov functional method. In Wang et al. (2015), Wang et al. proposed an adaptive control method by new definition of errors between drive system and response system and the synchronization speed can be adjusted by the gain of the adaptive controller. In Zhang and Shen (2014), Zhang et al. demonstrated exponential synchronization of memristor-based chaotic neural networks with both time-varying delays and general activation functions via periodically intermittent control.
Different from normal control method, pinning control aims to synchronize systems by controlling partial nodes instead of all nodes. In complex networks, pinning control has attracted a lot of attentions from researchers. In Wang, Shen, and Yin (2013), a pinning policy was proposed to synchronize MNN and coupled MNN exponentially under a mild topology condition. In Yang, Guo, and Wang (2015), Yang et al. proposed a pinning adaptive coupling method for robust synchronization of multiple MNNs with uncertain parameters. In Guo, Yang, and Wang (2016), Guo et al. introduced two types of distributed pinning control to achieve global exponential synchronization of multiple memristive neural networks in the mean square sense.
Even though some investigations about pinning control are obtained, most researchers focus on multiple MNNs and coupled MNNs. To the best of our knowledge, the synchronization of MNNs via pinning control has rarely been studied. In fact, coupling relationship also exists in MNNs whose state of each node can be affected by other nodes. Thus, it is possible to achieve synchronization of MNNs by controlling partial nodes instead of every node. Actually, the normal control policy can achieve better synchronization performance than pinning method. However, in some special situations, it is difficult to control every node and normal control method is useless. Thus, it is necessary to investigate pinning control policy which can be applied to solve this problem.
In this paper, we introduce a pinning control method for synchronization of MNNs with time-varying delays. The proposed pinning method has several advantages. First, the structures of pinning controllers are very simple and only a few parameters of controllers need to be determined. Second, it requires simple calculations. In previous works, LMIs are necessary for the parameters of controllers which need complex computations. Finally, the pinning control method can be improved to become a normal control policy which has better synchronization performance.
The rest of this paper is organized as follows. In Section 2, some preliminaries about MNNs are introduced. In Section 3, the pinning policy is proposed and some sufficient conditions for asymptotic and exponential synchronization of MNNs are discussed. Some improvements are also discussed in this section. In Section 4, a numerical example is presented to demonstrate the effectiveness of the obtained results and in Section 5, brief conclusions are given.
Section snippets
Problem description
MNNs can be implemented by large-scale integration circuits, as given in Fig. 1. In this paper, we consider a class of MNNs with time-varying delays which are described by the following differential equations: where is the voltage of the capacitor , denotes the external input to the th neuron which is usually a constant, corresponds to the transmission delay satisfying (
Pinning synchronization of MNNs with time-varying delays
In this section, some pinning synchronization criteria are discussed for synchronizing the systems (2), (4). Pinning control is a method that aims to control partial nodes to achieve synchronization goal for the entire network. Hence, the basic model of pinning control can be presented as follows: where is the number of the controlled nodes, is normal control input which can be applied in every node, and is appropriate pinning control input to achieve
Simulation studies
In this section, numerical simulations are given to verify the effectiveness of designed controller. Consider a three-dimensional MNN system as follows: where and and are given as follows:
Conclusions
In this paper, we apply pinning control on MNNs with time-varying delays for synchronization which is different from previous works Wang et al. (2013), Yang et al. (2015). A new pinning policy is proposed and some sufficient conditions for asymptotic and exponential synchronization of MNNs are also given. The proposed method can be used in MNNs which have nodes (). The pinning policy can be concluded as follows. First, choose appropriate nodes to control and the initial condition of the
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China under Grants 61533017, U1501251, 61374105, 61503377 and 61233001. The authors would like to thank anonymous reviewers for their valuable comments.
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