Centralized and Decentralized Data-Sampling Principles for Outer-Synchronization of Fractional-Order Neural Networks

This paper aims to investigate the outer-synchronization of fractional-order neural networks. Using centralized and decentralized data-sampling principles and the theory of fractional differential equations, sufficient criteria about outer-synchronization of the controlled fractional-order neural networks are derived for structure-dependent centralized data-sampling, state-dependent centralized data-sampling, and state-dependent decentralized data-sampling, respectively. A numerical example is also given to illustrate the superiority of theoretical results.


Introduction
Fractional operator has become visible in application domains [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].As the demanding performance expectations with uncertainty, fractional operator offers more degrees of freedom to designers to meet some predefined performance indexes.After gradually recognizing the importance of fractional operator, it is found that the description of fractional-order model is more accurate and totally different from that of the corresponding integer-order model.As a direct application, the characteristic of fractional-order model can be used to identify possible behavior of electrical signals from neurons.In physical implementation of neurodynamic systems, arbitrary order analog fractance circuit is most appropriate, which reveals profoundly the relationships among neural circuit elements [9][10][11].In that way, real neurodynamic systems should be addressed by fractional-order models.Fractional-order neurodynamic systems can better describe how action potentials in neurons are launched and spread.In addition, fractional-order neurodynamic systems possess infinite memory, and yet, integer-order neurodynamic systems are not of such feature [3][4][5][6][7][8][12][13][14][15].Therefore, fractional-order neurodynamic systems have the potential to accomplish what integerorder ones can not do.More feasible analysis methods and easy-to-use techniques to be deal with fractional-order neurodynamic systems are worth looking into.
As a coherent behavior within nonlinear systems, synchronization of nonlinear systems has attracted phenomenal worldwide attention.Many studies have shown that synchronization mechanism is a universal phenomenon and has a wide range of applications in engineering systems.Generally, two schemes for synchronization are frequently used: inner-synchronization and outer-synchronization.For innersynchronization, all nodes within a network will achieve a coherent behavior.However, for outer-synchronization, all individuals in two networks will achieve identical behaviors.In many application fields, outer-synchronization may seem practical [16][17][18][19][20][21][22][23].For example, in heuristic computational intelligence, it is known that outer-synchronization is rooted in brain-inspired computing from evolutionary strategies to cognitive tasks.Nevertheless, results focusing on outer-synchronization of complex control systems have seldom been reported [19].Control strategy for outersynchronization deserves more investigation.
Sampled-data control through only using the local information has recently generated significant research interest [24][25][26][27][28][29][30][31][32][33][34][35][36][37][38].Unlike continuous-time control, which requires the continuous communication data, sampled-data control is more appropriate under networked environment.For control systems, once we can give effective sampling policies and schedule, then the sampled-data control will reduce communication data and save energy dramatically.Thus, how to develop high-efficiency, heuristic information-based sampled-data control with the ultimate aim of maximizing the data collected is worth studying [38].However, relevant studies of the data-sampling strategy for control systems are still in early stage.
Motivated by the above discussions, in this paper, we introduce the centralized and decentralized data-sampling principles to achieve outer-synchronization between coupled fractional-order neural networks.The efficient allocation of the limited energy resources of centralized and decentralized data-sampling principles that maximizes the information value of the data collected is clearly a step forward.Meanwhile, to more efficiently design the sampling method, we merge the structure and state clusters through centralized and decentralized data-sampling principles and then select the best sampling time.On the basis of some analytical tools of fractional differential equations, a series of criteria on outersynchronization are derived.It should be noted that such criteria capture the information on sampling pattern and may have much wider application range.
The rest of the paper is organized as follows.In Section 2, we present the preliminaries and problem formulation.In Section 3, we state main results in detail.In Section 4, simulation example is illustrated.Finally, Section 5 concludes the paper.

Preliminaries and Problem Formulation
First, some preliminaries of fractional operator are given.
Fractional integral   (⋅) for H() with order  > 0 is described as where Γ(⋅) is Gamma function and  0 is the initial time.
One-parameter Mittag-Leffler function   (⋅) is described as where Γ(⋅) is Gamma function,  > 0, and  is a complex number.
For the centralized data-sampling principle, ( 4) is rewritten as where   is simple notion of  () with () = max{K :  K ≤ } and 0 =  0 <  1 < ⋅ ⋅ ⋅ <   < ⋅ ⋅ ⋅ is uniform for all the system states.Every neuron intersperses its state to its out-neighbors and receives the state information from its in-neighbors at the same time point   .
For the decentralized data-sampling principle, ( 4) is rewritten as where    is simple notion of   () with () = max{K :   K ≤ } and 0 =   0 <   1 < ⋅⋅⋅ <    < ⋅⋅⋅ is distributed for  ∈ {1, 2, . . ., }.Each neuron  pushes its state information to its out-neighbors at time    when it updates its state.It receives the information of in-neighbor state at time    when the neighbor neuron  updates its state.Now, we state definition and problem formulation.
In the following, we end this section with some notations that are needed later.

Main Results
For problem formulation in preceding section, in this section, we propose the corresponding control schemes for centralized data-sampling principle and decentralized datasampling principle, respectively.
To facilitate the narrative, we first address the control designs, then review, and analyze the theoretical results.
Remark 9.For the sampled-data control, how to choose the proper scheme with the ultimate aim of maximizing the data collected to control the system is challenging.For example, as revealed in [9,10], it is extremely difficult to design the sampling time point inherited from the sampled-data control strategy.However, according to Theorems 4-7, this situation can be effectively solved if the centralized and decentralized data-sampling principles are cleverly utilized.
Remark 10.For three control schemes in Theorems 4-7, these are just the type and level of points, not the merits of good points of difference.Theorem 4 is entirely focused around the centralized data-sampling principle via structure.Theorem 6 is concerned with the centralized data-sampling principle via state.Theorem 7 is to place emphasis on the decentralized data-sampling principle via state.
Remark 11.Note that the sampled-data control in Theorems 4-7 exerts only at the sampling time point, that is, every system state employs only its neighbors' information at   or    .Thus, compared with the continuous-time control strategy, the control schemes in Theorems 4-7 can effectively save the bandwidth and reduce the communication cost.Moreover, the results obtained here are the first ones on centralized and decentralized data-sampling principles for outer-synchronization of fractional-order neural networks.

Remark 12.
The key features of outer-synchronization in Theorems 4-7 are follows.(1) Each outer-synchronization scheme is closely related to the sampling time point.Once the sampling time point is given, the states of the controlled fractional-order neural networks will achieve outersynchronization. (2) Centralized data-sampling principle via structure makes full use of the characteristic of system itself, while centralized or decentralized data-sampling principle via state skillfully combines the feature of state measurement error.
Remark 13.The analytical methods for outer-synchronization in Theorems 4-7 are quite different from conventional complete synchronization, projective synchronization, phase synchronization, distributed synchronization, pinning synchronization, and cluster synchronization.
To select  1  according to Theorem 7, system (52) reaches outersynchronization. Figures 7 and 8 depict the dynamics of  1 () and V 1 (),  2 () and V 2 () in the triggering time points as Theorem 7, respectively.Figure 9 describes the release time points and release intervals.
Remark 14.In existing publications, there has been no theoretic criterion to achieve outer-synchronization of (52).In addition, using centralized or decentralized data-sampling principle to analyze and control fractional-order systems is also rare.
Release time points and release intervals   Remark 15.According to simulation analysis in Figures 1-9, there is no essential difference regarding outersynchronization performance in three control schemes as Theorems 4-7.By comparative analysis of Figures 3, 6, and 9, the release intervals via control scheme as Theorem 4 are relatively minor, and the triggering time points via control scheme as Theorem 7 are spread more thinly.

Concluding Remarks
In this paper, we show that outer-synchronization of fractional-order neural networks can be achieved by applying appropriate centralized and decentralized data-sampling principles.Such theoretical results improve and supplement some existing related results.The results obtained here are sufficient conditions for outer-synchronization of fractional-order neural networks and may remain room for improvement.Further extensions would be welcome: (1) outer-synchronization of fractional-order neural networks considering both conservativeness and complexity; (2) analyzing the outer-synchronization of fractional-order neural networks subject to time-delay; (3) analyzing the outersynchronization of fractional-order neural networks subject to stochastic disturbance.

Figure 6 :
Figure 6: The release time points and release intervals in the triggering mechanism as Theorem 6.