Stochastic Memristive Quaternion-Valued Neural Networks with Time Delays: An Analysis on Mean Square Exponential Input-to-State Stability

In this paper, we study the mean-square exponential input-to-state stability (exp-ISS) problem for a new class of neural network (NN) models, i.e., continuous-time stochastic memristive quaternion-valued neural networks (SMQVNNs) with time delays. Firstly, in order to overcome the difficulties posed by non-commutative quaternion multiplication, we decompose the original SMQVNNs into four real-valued models. Secondly, by constructing suitable Lyapunov functional and applying Itô’s formula, Dynkin’s formula as well as inequity techniques, we prove that the considered system model is mean-square exp-ISS. In comparison with the conventional research on stability, we derive a new mean-square exp-ISS criterion for SMQVNNs. The results obtained in this paper are the general case of previously known results in complex and real fields. Finally, a numerical example has been provided to show the effectiveness of the obtained theoretical results.


Introduction
In 1971, Chua proposed the theoretical idea of memristor [1], and its prototype was produced by the HP lab successfully in 2008 [2,3]. A memristor is the fourth fundamental element of an electrical circuit that can be used to construct a new artificial neural network. It has a tremendous potential to be utilized in synapsis for simulation of the human brain by replacing a resistor with a memristor [4][5][6]. In view of these characteristics, a new neural network (NN) model, namely, the memristive neural network (MNN) has been widely studied, and many theoretical papers regarding various dynamics of MNNs have been published in recent years [7][8][9][10][11][12][13]. From the real-world application perspective, time delays inherently arise in many practical systems including NNs. Indeed, time delays appear as the main cause of instability, compromising the system performance in real environments [10,[14][15][16][17][18]. into four RVNN models; (3) the main results of this paper are new and more general than those in the QVNN literature. This paper is structured as follows. We formally define the proposed model and analyze the new exp-ISS criteria in Sections 2 and 3, respectively. We present the numerical example and the associated simulation results in Section 4. In Section 5, we present the key conclusions and some suggestions for future research.

Mathematical Notations
The real field, complex field, and skew field of quaternion are denoted as R, C, and Q, respectively. Their n-dimension vector with elements of R, C and Q are denoted as R n , C n and Q n , while their n × n matrices with entries from R, C and Q are denoted as R n×n , C n×n and Q n×n , respectively. In addition, the space of a continuous function mapping ϕ from [−τ, 0] into Q n is denoted as C ([−τ, 0]; Q n ). The closure of the convex hull of Q n , which is formulated from quaternion numbers ∇ and , is denoted as co{∇, }. A class of essentially bounded function u from [0, ∞) to Q n with u ∞ = ess sup −τ≤s≤0 |u(s)| < ∞ is denoted as ∞ , while the family of all F 0 measurable is denoted as L 2 F 0 ([−τ, 0]; Q n ). Besides that, C ([−τ, 0]; Q n )-valued stochastic variables {ϕ(s) : −τ ≤ s ≤ 0} is in a way that 0 −τ E|ϕ(s)| 2 ds < ∞, in which the mathematical expectation operation pertaining to a probability measure P is denoted as E{·}. Superscripts * and T represent the complex conjugate transpose and matrix transposition, while i, j, k represent the imaginary units, respectively, and N = 1, 2, ..., n.

Quaternion Algebra
Firstly, we address the quaternion and its operating rules. We can express the quaternion, which consists of a real part along and three imaginary parts, as: where m R , m I , m J , m K ∈ R. The multiplication rules of Hamilton are satisfied by the imaginary roots i, j, k: The quaternion-valued function is denoted by m(t) = m R (t) + im I (t) + jm J (t) + km K (t) ∈ Q, where m R (t), m I (t), m J (t), m K (t) ∈ R. Let p = p R + ip I + jp J + kp K and q = q R + iq I + jq J + kq K are two quaternions, the addition and subtraction of p ± q are defined as We can express the product of pq with respect to the multiplication rules of Hamilton as: represents the norm of m, while |m| = √ mm * = (m R ) 2 + (m I ) 2 + (m J ) 2 + (m K ) 2 represents the modulus of m, in which the conjugate transpose of m is denoted as m * = m R (t) − im I (t) − jm J (t) − km K (t).

Problem Formation
In this section, we define a SMQVNN model as: or equivalently, for all x, z ∈ N, where a xz = max{â xz ,ǎ xz }, We assume that the NN model (6) can be separated into real and imaginary parts. Therefore, there exist measurable functionsã R Consider Model (7), its initial condition is for all m 1 , m 2 ∈ R and f s z (0) = 0, g s z (0) = 0, s = R, I, J, K.

Main Results
We derive the sufficient conditions that assure the mean-square exp-ISS with respect to the trivial solution of the SMQVNN model in Equation (2).
Proof. A Lyapunov functional is formulated as where Then, where Based on the Itô's formula, we have where Vm(t,m(t)) = ∂V (t,m(t)) ∂m 1 , ..., where the Itô's operator is L , such that According to A2 and A3, we have By using similar techniques, we can estimate the derivatives of L V 2 (t, m I (t)), L V 3 (t, m J (t)) and L V 4 (t, m K (t)) along the solution with respect to the model in Equation (7) as Using the well-known Young inequality, it follows that

Substituting the above inequalities into Equations (26)-(29), we have
According to Equations (14)-(21), we can easily obtain Now, similar to the study in [13], the Markov time or stopping time is defined as h := in f {s ≥ 0 : |m(s)| ≥h}. The Dynkin formula is then used to yield With respect to the monotone convergence theorem, we have the following whenh goes to ∞ Based on Equation (22), we have Combining Equations (37) and (38), we have Consequently, we can derive where which implies that the SMQVNN model in Equation (2) has a trivial solution, which is mean-square exp ISS. The proof is completed. 2s x c x ≥ λs x + Remark 5. In [39,40], the authors have studied the exp-ISS stability criteria for a class of stochastic RVNNs. Recently, in [13,41], the authors have extended the results proposed in [39,40], to the complex domain. The exp-ISS stability criteria for stochastic CVNNs has also been analyzed. Compared to the previous results, in this paper, we have generalized the results proposed in [13,[39][40][41] to the quaternion domain. Furthermore, the results obtained in previous studies [13,[39][40][41] can be seen as special cases of this paper.

Illustrative Examples
A simulation example is presented to ascertain the efficiency of the results.

Example 1.
A 2D SMQVNN model with a time-varying delay is considered as follows: where and c 1 = 6, c 2 = 6. The the activation functions and external input are as follows f(m(·)) = g(m(·)) = tanh m R (·) + i tanh m I (·) + j tanh m J (·) + k tanh m K (·) and U = [0.1 sin(t) + i sin(0.1t) + j sin(0.1t) + k sin(0.1t), 0.1 cos(t) + i cos(0.1t) + j cos(0.1t) + k cos(0.1t)] T . According to the above analysis, we have σ(t, m(t), From Theorem (1), it is clear that the model in Equation (54) is mean-square exp-ISS. Referring to the model in Equation (54) in the presence of inputs, its state responses with respect to the real and imaginary parts are plotted in Figures 1-4. On the other hand, the model in Equation (54) in the absence of the inputs has the state responses with respect to the real and imaginary parts shown in Figures 5-8. Therefore, it is clear that the model in Equation (54) is mean-square exponentially stable.

Summary
In this paper, we have studied the mean-square exponential input-to-state stability problem for a new class of SMQVNNs with time-varying delays. Firstly, we decomposed the original SMQVNNs into four real-valued models, in order to avoid the difficulties posed by non-commutative quaternion multiplication. Secondly, by constructing suitable Lyapunov functional and applying Itô's formula, Dynkin's formula as well as inequity techniques, several new sufficient conditions have been obtained to guarantee that the considered system model is mean-square exp-ISS. The results obtained in this paper are the general case of previously known results in complex and real fields. Finally, a numerical example shows the effectiveness of the obtained theoretical results.
Based on the proposed method in this paper, it is possible to analyze different QVNN models. The proposed approach can be applied to the analysis of stochastic QVNN models for the discrete-time case. In future studies, the stability and synchronization analysis for discrete-time stochastic QVNN models will also be investigated. The results will be useful for the dynamical analysis of discrete-time stochastic QVNN models.