Global asymptotic stability of fractional-order competitive neural networks with multiple time-varying-delay links

Highlights

• Multiple time-varying delay links are introduced into fractional-order competitive neural networks.

• An integral inequality associated with every upper bound of each time-varying delay is proposed.

• Based on integral inequality, a generalized fractional-order Halanay inequality is obtained.

• The stability criterion is dependent on the topological structure of networks.

Abstract

Competitive neural networks have become increasingly popular since this kind of neural networks can better describe the dynamics of cortical cognitive maps with unsupervised synaptic modifications. In this paper, we first propose fractional-order competitive neural networks with multiple time-varying-delay links and explore the global asymptotic stability of this class of neural networks. A novel and generalized integral inequality related to every upper bound of each time-varying delay is given. Moreover, based on Lyapunov method and graph theory, we obtain some sufficient conditions with the help of this integral inequality to guarantee the global asymptotic stability. The theoretical results offer a new perspective to show the close relationship between the stability criterion and the topological structure of networks. Finally, an illustrative numerical example is given to demonstrate the feasibility and effectiveness of the theoretical results.

Introduction

In recent years, various kinds of neural networks have been extensively studied such as Hopfield-type neural networks, recurrent-type neural networks, Cohen-Grossberg neural networks and so on [1], [2], [3], [4]. It is essential to investigate neural networks intensively as a consequence of wide applications in many fields such as pattern recognition, signal processing and combinatorial optimization, see [5]. It is worth noting that only one time scale is taken into consideration in the aforementioned neural networks, in other words, there is only one type of state variables of the neurons. In 1983, a class of competitive neural networks (CNNs) was first studied [6]. Afterwards, Meyer-Baese et al. [7] generalized two different time scales on CNNs which usually comprise of two classes of state variables including long-term memory (LTM) and short-term memory (STM). LTM depicts the slow unsupervised synaptic modifications while STM describes the rapidly changing dynamical behaviors of neurons. Besides, as we know, time delays have occurred in almost neural networks because of the signal transmission lag and the finite switch speed of amplifiers in the circuit implementations. Distinguished from constant delays, time-varying delays are more general owing to the property of evolution along with time [8], [9], [10], [11], [12]. Therefore, CNNs with time-varying delays have received increasing concern in the past few years [13], [14], [15], [16], [17], [18]. The general CNNs with time-varying delay can be expressed as follows:

$$\{\begin{array}{c}{\displaystyle {\dot{r}}_j\left(t\right)=-c_j{r}_j\left(t\right)+\sum _{k=1}^m{u}_{jk}{h}_k\left(r_k\left(t\right)\right)+\sum _{k=1}^m{v}_{jk}{h}_k\left(r_k(t-\tau \left(t\right))\right)+w_j\sum _{i=1}^b{s}_{ji}\left(t\right){\epsilon }_i+I_j,}\hfill \\ {\dot{s}}_{ji}\left(t\right)=-d_j{s}_{ji}\left(t\right)+{\epsilon }_i{h}_j\left(r_j\left(t\right)\right),i\in \mathbb{B},\hfill \end{array}j\in \mathbb{M},$$

where $\mathbb{M}=\{1,2,\dots ,m\},$ $\mathbb{B}=\{1,2,\dots ,b\},$ $r_j\left(t\right)\in \mathcal{R}$ stands for the neural current activity level, $s_{ji}\left(t\right)\in \mathcal{R}$ denotes the synaptic efficiency and function h_k represents the output of neurons; c_j > 0 represents the time constant of neuron, ε_i > 0 indicates the constant external stimulus, w_j > 0 is the strength of the external stimulus, d_j > 0 denotes disposable scaling constant, I_j represents the constant input and τ(t) expresses the coupling time-varying delay satisfying 0 ≤ τ(t) ≤ τ; u_jk ≥ 0, v_jk ≥ 0 denote the coupling strength and (u_jk)_m × m, (v_jk)_m × m reflect the topological structure of the subnetworks.

It is notable that the above mentioned papers about CNNs are on the basic of integer calculus. Fractional calculus, as a typical mathematical notion with a long history, is the promotion of integer calculus and derivation and integration of arbitrary non-integer order. Since fractional operator is nonlocal, it becomes an excellent tool to describe some materials and processes with memory and hereditary properties. Regarding the benefits, fractional calculus has generated lots of results in applications including heat conduction, epidemic modelling and colored noise, therefore fruitful theoretical research results have been obtained as well, see [19], [20], [21], [22], [23] for example. Note that when incorporating fractional calculus into CNNs, common capacitor from integer-order CNNs is changed into fractional capacitor so that the accuracy and memory property of CNNs are largely improved. Therefore, in recent few years, some scholars have extended CNNs to fractional-order case and made deep dynamical analysis of fractional-order CNNs [24], [25], [26]. For example, in [24], Liu et al. discussed the coexistence and dynamical behaviors of multiple equilibrium points for fractional-order CNNs with Gaussian activation functions. Stability and synchronization criteria for fractional-order CNNs with time delays were derived in [26].

The existing papers have made sufficient explorations in a great many aspects of dynamical behaviors in CNNs whereas at most two links are the prerequisite for these results. It is far beyond ignored that most networks in reality exist with multiple links such as biological neural networks, cellular networks, ecological networks and so on. Neural networks with multiple links express more than one type of connection between any pair of diverse nodes and each line endowed with its exclusive property [27], [28], [29]. Taking biological neural networks for instance, biological neurons, as a kind of cells in the nervous system, utilize synapses networks of the neurons to carry out communications among neural cells to actualize processing and transmission of information, which are characterized by multiple connectivity. In [30], it is noted that there are diverse connections in accordance with different neural structures for biological neural networks. Therefore, it is of great necessity to introduce multiple links when modelling CNNs. In this paper, we introduce the following fractional-order CNNs with multiple time-varying-delay links built on a digraph $\mathcal{G},$ which are described as follows:

$$\{\begin{array}{c}{\displaystyle {}_0{\mathrm{D}}_t^{\lambda }{r}_j\left(t\right)=-c_j{r}_j\left(t\right)+\sum _{k=1}^m{u}_{jk}{h}_k\left(r_k\left(t\right)\right)+\sum _{n=1}^l\sum _{k=1}^m{v}_{jk}^{\left[n\right]}{h}_k\left(r_k(t-{\tau }_n\left(t\right))\right)+w_j\sum _{i=1}^b{s}_{ji}\left(t\right){\epsilon }_i+I_j,}\hfill \\ {}_0{\mathrm{D}}_t^{\lambda }{s}_{ji}\left(t\right)=-d_j{s}_{ji}\left(t\right)+{\epsilon }_i{h}_j\left(r_j\left(t\right)\right),i\in \mathbb{B},\hfill \end{array}j\in \mathbb{M},$$

where ${}_0{\mathrm{D}}_t^{\lambda }$ denotes Caputo’s fractional derivative with order λ ∈ (0, 1), τ_n(t) expresses the coupling time-varying delay satisfying 0 ≤ τ_n(t) ≤ τ_n, $n=1,2,\dots ,l$. Besides, u_jk ≥ 0, $v_{jk}^{\left[n\right]}\ge 0$ denote the coupling strength and (u_jk)_m × m, ${\left(v_{jk}^{\left[n\right]}\right)}_{m\times m}$ reflect the topological structure of the 0th subnetwork and the nth subnetwork, respectively. Other explanatory notes for parameters are the same with system (1).

Based on the discussions above, we provide a framework for stability of fractional-order CNNs with multiple time-varying-delay links. As is known to all, Lyapunov method is a great and important technique to deal with dynamical behaviors of complex networks including neural networks. As is known to all, Lyapunov method is a great and important technique to deal with dynamical behaviors of complex networks including neural networks [26], [31], [32], [33], [34]. For example, in [31], based on Lyapunov method, the global stability of fractional-order Hopfield neural networks with time delay was discussed. Notwithstanding, constructing an appropriate Lyapunov function cannot be fulfilled with ease. Inspired by [35], with the help of Kirchhoff’s Matrix Tree Theorem in graph theory, we can obtain a Lyapunov function associated with the topological structure of networks, which avoids the difficulty of constructing a Lyaounov function directly. In addition, we give a novel and generalized integral inequality associated with every upper bound of each time-varying delay. Based on Lyapunov method and graph theory, a stability criterion is attained with the help of this integral inequality. To clearly illustrate the feasibility of our theoretical results, a numerical example is also presented. Moreover, the noteworthy contributions of this paper can be stated as follows:

• We generalize integer-order CNNs into fractional-order CNNs with multiple time-varying-delay links. In contrast to [21] and [25] which consider neural networks with a single coupling time-varying delay, the model we study can reflect the multiple time-varying-delay links among neurons more realistically.

• A novel and generalized integral inequality associated with every upper bound of each time-varying delay is proposed. Based on this inequality, we can obtain a generalized fractional-order Halanay inequality which is related to every upper bound of each time delay.

• The theoretical results offer a new perspective to show the close relationship between the stability criterion and the topological structure of networks.

  The structure of the paper is outlined as follows. In Section 2, preliminary notes and problem statement are given. Main results are provided in Section 3. And a numerical example is presented in Section 4. Finally, our conclusion is drawn in Section 5.