Projective synchronization for quaternion-valued memristor-based neural networks under time-varying delays

Notations: Throughout this paper, $\mathbb{R}$, $\mathbb{C}$, and $\mathbb{Q}$ denote the real, complex, and quaternion fields, respectively. The notation adheres to standard mathematical conventions.

1.   Introduction

The memristor, a pioneering circuit component that encapsulates the interplay between magnetic flux and electric charge, made its debut in scientific literature through the visionary work of Chua ^[1] in 1971. This innovative element distinguishes itself from traditional circuit components due to its nonlinear resistance and inherent memory capabilities. Its mnemonic attribute bears a striking resemblance to the synaptic plasticity observed in the neural connections of the human brain. Leveraging this distinctive feature, the memristor has transcended its role as an electrical resistor to become a cornerstone in simulating the cognitive functions of the human brain. Consequently, the exploration of memristor-based neural networks (MBNNs) has emerged as a vibrant field of study, with researchers delving into the intricate dynamics that these networks exhibit ^[2,3,4]. MBNNs are characterized as state-dependent dynamical systems, with coefficients that are inherently linked to their instantaneous state. The discontinuous nature of these systems can give rise to a spectrum of complex nonlinear phenomena, including but not limited to chaos, oscillations, and instability. The investigation of these dynamical traits is not merely an academic pursuit; it holds profound significance in both theoretical exploration and practical applications, paving the way for advancements in neuromorphic engineering and computational neuroscience.

The quaternion, a mathematical construct first articulated by the Irish mathematician Sir William Rowan Hamilton in 1843, stands as an innovative extension of the real and complex number systems. Unlike their simpler counterparts, quaternions introduce a non-commutative multiplication, a feature that has historically posed challenges and led to a period of relative dormancy in their study. However, in recent years, there has been a resurgence of interest and a broadening scope of applications for quaternions. They have demonstrated remarkable utility across a spectrum of disciplines, including artificial intelligence ^[5], image processing ^[6], quantum mechanics ^[7], and aerospace technology ^[8]. These applications have not only reinvigorated the study of quaternions but also unveiled their potential to address complex problems in a multidimensional context. Especially in image processing, the nuanced representational power of quaternions has been harnessed. They are used to encode the three primary color channels within the imaginary components of a quaternion, while the real part is often reserved for the alpha channel or other metadata. This elegant mapping of color images to pure quaternions has opened new avenues for the representation and manipulation of visual data.

Quaternion-valued neural networks (QVNNs) distinguish themselves from their complex-numbered counterparts by employing quaternions in every aspect of their architecture—states, connection weights, and activation functions. This holistic approach to quaternion integration endows QVNNs with a unique capacity for handling multidimensional data representations. In recent scholarly discourse, a surge of interest has been directed towards the dynamics of QVNNs, as evidenced by a burgeoning body of literature ^{[9,10,11,12,13,14]}. Notably, the exploration of robust stability within the fractional-order realm of QVNNs has garnered significant attention, as demonstrated by the contributions of ^[9]. Additionally, innovative control methodologies, such as sampled-data approaches, have been applied to stabilize QVNNs, as explored in ^[10]. The investigation into the robust stability of these networks continues to evolve, with offering fresh insights into the fractional-order context ^[13]. Furthermore, the stability analysis of quaternion-valued memristive neural networks has been enriched by the application of Lagrangian mechanics, as discussed in ^[14], and so on ^{[15,16,17,18,19]}. These studies collectively contribute to a deeper understanding of the intricate dynamics that govern QVNNs, paving the way for advancements in network stability and synchronization.

The recent research on quaternion-valued memristive neural networks (QVMNNs) has attempted to understand their periodic solutions, as highlighted in ^[15]. This work has been complemented by an examination of dissipativity in neutral-type memristor-based networks, which provides a fresh perspective on stability and energy dynamics, as discussed in ^[16]. Moreover, the exploration of finite-time stabilization through implicit function methods, as introduced in ^[17], has opened new avenues for the rapid and reliable control of these networks, underscoring the growing sophistication in the field of QVNNs.

Synchronization, a fundamental and pivotal phenomenon in the realm of neural networks, plays a crucial role in managing and orchestrating the inherent chaotic dynamics that are often observed in natural systems. It serves as a powerful tool for harnessing order from chaos, offering a mechanism to regulate and predict the behavior of complex neural systems. Through synchronization, we can explore unknown dynamical systems from known ones. So far, the synchronization of neural networks includes quasi-uniform synchronization ^[20,21,22], anti-synchronization ^[23,24], finite-time synchronization ^[25,26,27], projection synchronization ^[28,29], exponential synchronization ^[30,31], and global Mittag-Leffler synchronization ^[32,33], and others ^[34,35].

Among the spectrum of synchronization phenomena, projection synchronization emerges as an inclusive and versatile approach within the domain of neural network systems. This method, underpinned by proportional dynamics, facilitates accelerated communication pathways, a feature that sets it apart from other synchronization modalities ^[36,37,38]. In this work, we extend the discourse to the projective synchronization of quaternion-valued memristor-based neural networks, a topic that gains relevance amidst time-varying delays. Our exploration is anchored in the following pivotal contributions:

(ⅰ) QVMNNs offer distinctive benefits over traditional real- and complex-valued neural networks, particularly in their adept handling of multi-dimensional data through low-dimensional constructs and enhanced computational efficiency.

(ⅱ) The synchronization criteria established within this study are not only capable of orchestrating complete synchronization and anti-synchronization but also of embracing the general projection synchronization paradigm. These findings are posited as universally applicable and representative of the broader scope of neural network synchronization.

(ⅲ) By designing a novel controller and using Lyapunov function and Young's inequality, some new synchronization conditions for projection synchronization of quaternion-valued memristor-based neural networks are obtained.

2.   Problem description and preliminaries

The quaternion constitutes a realm of hypercomplex numbers, encapsulating a singular real component alongside three imaginary elements, thereby extending the numerical system beyond the conventional real and complex planes. The quaternion $m\in \mathbb{Q}$ can be described as

where $m^R, m^I, m^J, m^K \in \mathbb{R}$, the imaginary parts $i, \; j, \; k$ obey the Hamilton rule:

Remark 1. Quaternions, a class of hypercomplex numbers, diverge from the algebraic properties of real and complex numbers. Unlike their commutative counterparts, the multiplication of any two quaternions, denoted as $m, \; n \in \mathbb{Q}$, does not necessarily adhere to the associative law $mn = nm$. This departure signifies that several principles governing real and complex number systems are inapplicable to quaternions. Consequently, it mandates the innovation of novel methodologies and theoretical frameworks to effectively harness and comprehend the intricacies of quaternion arithmetic.

For the quaternion $m$ and $n$, where $n = n^R+n^Ii+n^Jj+n^Kk$, then we can denote $m+n$ as follows:

By Hamilton rule, we can denote $mn$ as

The modulus of $m$ is written as

Furthermore, for $m = (m_1, m_2, \ldots, m_n)^T$, $\Vert m \Vert = \sum_ {i = 1} ^n |m_i|$ denotes the norm of $m$.

In this study, we delve into the dynamics of a memristor-driven neural network enriched with quaternion-valued parameters and subject to time-varying delays. The system's evolution is intricately captured by a set of differential equations that account for these temporal lags.

with each neuron's state vector represented by the components $m(t) = (m_1(t), m_2(t), \cdots, m_n(t))^T \in \mathbb{Q}^n$, $q = 1, 2, \cdots, n$, and $m_q(t)\in \mathbb{Q}$ corresponds to the individual neuron's state. The positive self-feedback coefficient is denoted by $c_q > 0$, and the quaternion-valued connection weights, stemming from memristor dynamics, are given by $b_{qs}(m_q(t))$ and $d_{qs}(m_q(t))$. The vector $f_s(m_s(t)) = (f_1(m_1(t)), f_2(m_2(t)), \cdots f_n(m_n(t)))$ encapsulates the activation functions that govern the neurons' firing patterns. The external stimuli to the network are captured by the input vector $I_q = (I_1, I_2, \cdots, I_n)^T \in \mathbb{Q}^n$. Additionally, the network's signal transmission is subject to delays characterized by $\tau(t)$, which are constrained to be non-negative and less than a maximum delay, satisfying $0\leq \tau(t) < \tau$. For the initial setup of the system (2.1), we select an initial condition that is continuously differentiable over the interval $[-\tau, \; 0]$, This is mathematically expressed as $m_q(s) = \psi_q(s) = (\psi_1(s), \psi_2(s), \cdots, \psi_n(s))\in C^{(1)}([-\tau, \; 0], \mathbb{Q}^n), -\tau\leq s\leq0$, laying down a well-defined starting point for the system's trajectory.

Incorporating the framework of differential inclusion and set-valued mappings, coupled with our preceding discourse, the representation of system (2.1) is delineated as:

where $b_{qs}^- = min\{\acute b_{qs}, \grave b_{qs}\}, b_{qs}^+ = max\{\acute b_{qs}, \grave b_{qs}\}$, $d_{qs}^- = min\{\acute d_{qs}, \grave d_{qs}\}, d_{qs}^+ = max\{\acute d_{qs}, \grave d_{qs}\}$. The essence of differential inclusion encapsulates the existence of a collection of differential equations where the terms $b_{qs}(t)\in co(b_{qs}^-, b_{qs}^+)$, $d_{qs}(t)\in co(d_{qs}^-, d_{qs}^+)$, such that

Assumption 1. Let $m\; = \; m^R\; +\; m^Ii\; +\; m^Jj\; +\; m^Kk$, where $m^R, \; m^I, \; m^J, \; m^K\in \mathbb{R}$. $f_s(m)$ can be decomposed into its real and imaginary constituents, denoted by $f_s(m)\; = \; f_s^R(m^R)\; +\; f_s^I(m^I)i\; +\; f_s^J(m^J)j\; +\; f_s^K(m^K)k$.

Assumption 2. For any $m_q^\epsilon(t), n_q^\epsilon(t)\in \mathbb{R}^n$, we identify the existence of certain positive constants $\epsilon = R, \; I, \; J, \; K, \; q = 1, \; 2, \; \cdots, \; n$, such that

Under Assumption 1, we divide the network (2.1) into one real part and three imaginary parts, respectively.

Based on the characteristics of the memristor, the quaternion-valued memristive connective weights are defined as

where the switching jump $T_{q} > 0$.

Consider the system (2.1) as the drive system; then, the response system is given as

where $q = 1, 2, \cdots, n$; $n(t) = (n_1(t), n_2(t), \cdots, n_n(t))^T \in \mathbb{Q}^n$. $n_q(t)\in \mathbb{Q}$ stand for the state vector of the neuron. The initial condition of system (2.4) is chosen to be $n_q(s) = \phi_q(s) = (\phi_1(s), \; \phi_2(s), \; \cdots, \; \phi_n(s))\in \mathbb{C}^{(1)}([-\tau, \; 0], \; \mathbb{Q}^n), \; -\tau\leq s\leq0$. $u_q(t)$ is the controller.

Based on the theory of differential inclusion set-valued map, it yields from Eq (2.8) that

Differential inclusion means that there exist $b_{qs}^*(t)\in co(b_{qs}^-, b_{qs}^+)$, $d_{qs}^*(t)\in co(d_{qs}^-, d_{qs}^+)$ such that

Similarly, we divide the network (2.10) into one real part and three imaginary parts, respectively

Let $e(t) = (e_1(t), e_2(t), \cdots, e_n(t))$ be the synchronization error. The synchronization error between the drive system (2.2) and the response system (2.9) is defined as $e_q(t) = n_q(t)-\beta m_q(t)$, expressed as one real part and three imaginary parts

with the initial value $\phi_q(s)-\psi_q(s)$, $-\tau \leq s \leq0$.

In order to synchronize the drive system and the response system, we choose the following controller

Let $e_q(t)\; = \; e_q^R(t)\; +\; e_q^I(t)i\; +\; e_q^J(t)j\; +\; e_q^K(t)k$; then, according to the controller (2.16), the error system (2.15) can be separated into four real parts as below

where $\tilde f_s^\epsilon(e_s^\epsilon(t))\; = \; f_s(n_s^\epsilon(t))\; -\; f_s(\beta m_s^\epsilon(t))$, $\tilde f_s^\epsilon(e_s^\epsilon(t-\tau(t))) = f_s(n_s^\epsilon(t-\tau(t)))-f_s(\beta m_s^\epsilon(t-\tau(t)))$, $\epsilon = R, \; I, \; J, \; K$.

The following notations will be used:

$| b_{qs}^R| = \mathop {\sup }\limits_{ t \geq 0} |b_{qs}^{*R}(t)|$, $| b_{qs}^I| = \mathop {\sup }\limits_{ t \geq 0} |b_{qs}^{*I}(t)|$, $| b_{qs}^J| = \mathop {\sup }\limits_{ t \geq 0} |b_{qs}^{*J}(t)|$, $| b_{qs}^K| = \mathop {\sup }\limits_{ t \geq 0} |b_{qs}^{*K}(t)|$,

$| d_{qs}^R| = \mathop {\sup }\limits_{ t \geq 0} |d_{qs}^{*R}(t)|$, $| d_{qs}^I| = \mathop {\sup }\limits_{ t \geq 0} |d_{qs}^{*I}(t)|$, $| d_{qs}^J| = \mathop {\sup }\limits_{ t \geq 0} |d_{qs}^{*J}(t)|$, $| d_{qs}^K| = \mathop {\sup }\limits_{ t \geq 0} |d_{qs}^{*K}(t)|$.

Before deriving the result, the definitions and lemmas are given to facilitate the subsequent derivation.

Definition 2.1. ^[38] The driving network (2.1) and the response network (2.8) are said to achieve projection synchronization if

where $\beta \in \mathbb{R}$ is a nonzero constant.

Remark 2. When the projection factor $\beta = 1$, complete synchronization is achieved. When the projection factor $\beta = -1$, anti-synchronization is obtained.

Lemma 2.1. ^[28] Let $m > 0$, $n > 0$, $r > 1$ and $\frac{1}{r}+ \frac{1}{s} = 1$. Then, the following inequality holds

Lemma 2.2. ^[39] Suppose that function $V(t)$ is non-negative when $t\in (-\tau, \infty)$ and satisfies the following inequality

where $a$ and $b$ are positive constants with $a > b$. Then,

where $r$ is the unique positive solution of the following equation

3.   Main results

Theorem 3.1. Under the controller (2.16), if Assumptions 1 and 2 hold, and

Then the projection synchronization of quaternion-valued memristor-based neural networks (2.1) and (2.8) is obtained, where $\lambda = min\{\lambda_1, \; \lambda_2, \; \lambda_3, \; \lambda_4\}$, $\zeta = max\{\zeta_1, \; \zeta_2, \; \zeta_3, \; \zeta_4\}$ and

Proof. Considering the following Lyapunov function

Computing the time derivative of $V(e(t))$ along the trajectory (2.17)–(2.20), from Lemma 2.1, we have

According to Assumption 2 and the notations defined previously, we can get

Then, according to Lemma 2.2, we have

Therefore, one has

Then, we have

We suppose that

Then,

where $\lambda = min\{\lambda_1, \lambda_2, \lambda_3, \lambda_4\}$, $\zeta = max\{\zeta_1, \zeta_2, \zeta_3, \zeta_4\}$.

Therefore, according to Lemma 2.4 and Theorem 3.1, the projection synchronization between drive system (2.1) and response system (2.8) can be achieved.

Remark 3. In recent years, the research of real-valued fractional-order neural networks and complex-valued fractional-order neural networks in projection synchronization^{[28,29,36,37]} has also achieved excellent results. Compared with previous studies, QVNN is superior in handling multi-dimensional problems. Therefore, our results are more general.

4.   Numerical example

Considering the two dimensional quaternion-valued memristor-based neural networks model of Eq (2.1) with the memristive connection weights are

The response system of (2.8) is

Take the time delay $\tau(t) = 0.75-0.25cos(t)$ such that $\tau = 1$ and the activation function $f(m_q(t)) = \; 0.23*(|m_q^R(t)+1|-|m_q^R(t)-1|)+\; 0.23*(|m_q^I(t)+1|-|m_q^I(t)-1|)i +0.23*(|m_q^J(t)+1|-|m_q^J(t)-1|)j+0.23*(|m_q^K(t)+1|-|m_q^K(t)-1|)k$ $(q = 1, 2)$, and external inputs $I_1$ = $I_2$ = 0. According to the Assumption 2, we can get $l_q^\varepsilon = 0.46$, where $q = 1, 2$. The control gain parameter is taken as $k_1 = k_2 = 40$ and the connection weights are taken as $c_1 = c_2 = 1$. Then, the trajectories of error systems with controllers and without controllers are obtained.

Besides, we take $r = 2$. According to the above parameters, we can directly calculate the direct result of the condition in Theorem 3.1: $\lambda_1 = \lambda_2 = \lambda_3 = \lambda_4 = 53.526$ and $\zeta_1 = \zeta_2 = \zeta_3 = \zeta_4 = 10.166$. Then, we can obtain $\lambda = 53.526 > \zeta = 10.166$.

The above result satisfies the condition in Theorem 3.1, so systems (2.1) and (2.8) achieve projective synchronization. Figures 1 and 2 show the synchronization error curves $e_q$ $(q = 1, 2)$ for $\beta = 0.8$ without the controller. Figures 3 and 4 show the synchronization error curves $e_q$ $(q = 1, 2)$ for $\beta = 0.8$ with the controller (2.16). Figures 5 and 6 show the synchronization error curves $e_q$ $(q = 1, 2)$ for $\beta = -1$ with the controller (2.16). Figures 7 and 8 show the synchronization error curves $e_q$ $(q = 1, 2)$ for $\beta = 2$ with the controller (2.16). From the above simulation results, we know that the derive system (2.1) and the response system (2.8) are synchronized, which verifies the effectiveness of Theorem 3.1.

5.   Conclusions

This study delves into the dynamics of projective synchronization within the realm of quaternion-valued memristor-based neural networks, which are subject to time-varying delays. Employing the theoretical underpinnings of set-valued mappings and differential inclusion, we formulate a hybrid control approach to dissect the projection synchronization dilemma of the network. By harnessing the stability assurances of a Lyapunov function and the quantitative bounds provided by Young's inequality, we formulate a novel criterion for synchronization. This leads to the achievement of projective synchronization in the context of the aforementioned neural networks. The efficacy and practicality of our proposed strategy are substantiated through rigorous numerical simulations.

Author contribution

Jun Guo, Yanchao Shi and Yanzhao Cheng wrote the main manuscript text. All authors reviewed the manuscript.

Acknowledgments

Supported by the National Natural Science Foundation of China under Grant 61703354; Key Laboratory of Numerical Simulation of Sichuan Provincial Universities KLNS-2023SZFZ002; the Scientific Research Foundation of Chengdu University of Information Technology KYQN202324.

Use of AI tools declaration

The authors declare that they have not used Artificial Intelligence (AI) tools in the creation of this article.

Conflict of interest

The authors declare there is no conflict of interest.