Global Synchronization and Anti-Synchronization of Fractional-Order Complex-Valued Gene Regulatory Networks With Time-Varying Delays

This paper presents the analytical and numerical investigation on the global synchronization and anti-synchronization for a class of drive-response systems of fractional-order complex-valued gene regulatory networks with time-varying delays (DFGRNs). In our design, two kinds of adaptive feedback controllers are used to synchronize and anti-synchronize the proposed drive-response systems, and some sufficient conditions on the global asymptotical synchronization and anti-synchronization are given with the methods of the fractional Lyapunov-like functions and the fractional-order inequalities. In the numerical simulations, two minimum “estimated time”, <inline-formula> <tex-math notation="LaTeX">$T_{1}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$T_{2}$ </tex-math></inline-formula>, are computed to achieve the synchronization and anti-synchronization. We find that <inline-formula> <tex-math notation="LaTeX">$T_{1}$ </tex-math></inline-formula> and <inline-formula> <tex-math notation="LaTeX">$T_{2}$ </tex-math></inline-formula> increase with the decreasing of the fractional order of DFGRNs.


I. INTRODUCTION
Genetic regulatory networks (GRNs) are fundamental and important biological networks that describe the interaction functions in gene expressions between DNAs, RNAs, proteins and small molecules in an organism [1]- [5]. Various GRNs models, such as Boolean networks [2], Bayesian networks [5], Petri networks [6], differential equation models [7]- [9], have been proposed by researchers over a period of time. And the researches on GRNs not only provide a powerful tool for elucidating the gene regulation processes in living organisms, but also contribute to the diagnosis of cancers, diabetes and other complex diseases [10]- [16].
It is worthy to note that, the differential equation models involving integer-order type and fractional-order type, where the state variables usually denote concentrations of messenger ribonucleic acids (mRNAs), proteins and other small molecules, is one of important GRNs model and The associate editor coordinating the review of this manuscript and approving it for publication was Rathinasamy Sakthivel . widely applied to describe the gene regulatory process [10], [11], [17]. Particularly, in [18]- [20], the authors pointed out that fractional derivatives provide an excellent tool for the description of memory and hereditary properties of various processes. Taking into account these facts, the incorporation of a memory term into a neural network model is an extremely important improvement [21]. Subsequently, the dynamics of the fractional-order GRNs becomes a hot topic and some remarkable results were reported in [7], [12], [16], [22]. Ji et al. [22] constructed fractional-order GRNs and demonstrated that the fractional-order model has stronger data approximation ability, which is more suitable for modeling gene regulatory mechanism.
Since the pioneer work of Perora and Carroll [23], synchronization and anti-synchronization as one of the most important dynamic behaviors has attracted increasing attentions and many significant results were derived [21], [24]- [35]. Different types of synchronization and anti-synchronization, such as adaptive synchronization [27], adaptive antisynchronization [28], finite-time synchronization [33], [36], VOLUME 8, 2020 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ finite-time anti-synchronization [34], Mittag-Leffler synchronization [30], [31], [37] and projective synchronization [21], have been widely investigated. Song et al. [25] considered the synchronization of fractional-order Lorenz chaotic systems and Chen chaotic systems with time delay, and designed the compensation controllers and optimal controllers. Chen et al. [26] studied the synchronization of memristor-based BAM neural networks with delays and realized asymptotic synchronization and exponential synchronization by designing two kinds of adaptive feedback controllers, respectively. Bao et al. [27] investigated the synchronization for fractional-order complex-valued neural networks with constant time delay and obtained the sufficient conditions of synchronization by using linear delay feedback control. And the main methods analyzing the synchronization and anti-synchronization of fractional-order dynamic system include, but are not limited to, direct quaternion approach [37], fractional-order comparison theory [18], fractional-order inequality [27] and fractional Lyapunov function approach [11], [18], [31]. The fractional Lyapunov method is a powerful tool for analyzing the synchronization and anti-synchronization of fractional-order dynamic system, which can be verified easily without solving the system. In recent years, the investigation on the synchronization of GRNs, which may be helpful to explore the biological rhythm and internal mechanisms at the molecular and cellular levels, has attracted attentions of researchers [11], [17]. Jiang et al. [17] considered the finite-time synchronization of integer-order GRNs without time delays, and established some sufficient conditions for finite-time stochastic synchronization by designing a continuous finite-time controller. Due to slow biochemical processes such as gene transcription, translation and transportation, time delays are omnipresent in GRNs [4], [9], [11], [14], [15], [38]- [40]. Qiao et al. [11] established some sufficient conditions of finite-time synchronization for fractional-order GRNs with constant time delay by designing respectively the state feedback controller and the adaptive controller. As time delays often change with time and its precise measurement is difficult in the real biological networks [41], it is therefore better practical significance to consider the time-varying delays, rather than the constant delays, for the GRNs [9], [40].
However, according to our knowledge, few researches have been given to the synchronization and anti-synchronization of fractional-order complex-valued GRNs with time-varying delays.
From above discussions, we will focus on the global asymptotical synchronization and anti-synchronization for a class of complex-valued FGRNs with time-varying delays. The rest of this paper is organized as follows. The preliminaries and the model description are provided in Section II. Section III proposes some sufficient criteria on global asymptotical synchronization and anti-synchronization for the DFGRNs. Section IV gives some numerical simulations to support our findings. And finally, Section V presents a brief discussion and the summary around the main results.

II. PROBLEM DESCRIPTION AND PRELIMINARIES
In general, three definitions of the fractional-order derivatives, the Grunwald-Letnikov derivative, the Riemann-Liouville derivative and the Caputo derivative, are mentioned. Rather than the other two definitions, the initial conditions for the Caputo fractional derivative can be determined only by the integer derivative, and the Laplace transform can be performed more concisely, it is therefore widely used in differential equation models. In particular, the Caputo fractional derivative is more suitable for the GRNs due to its more accurate description of the memory and hereditary characteristics of various materials and processes [37], [42]. Therefore, we adopt the Caputo fractional-order derivative.
Definition 1 [42]: The fractional integral of order q for a function f (t) is defined as where t ≥ a and n is a positive integer such that n − 1 < q < n.
The following continuous fractional-order complex-valued GRNs with time-varying delays are considered as the drive system: where m(t) = [m 1 (t), · · · , m n (t)] T ∈ C n , p(t) = [p 1 (t), · · · , p n (t)] T ∈ C n , t ≥ 0, A = diag{a 1 , · · · , a n } ∈ R n×n , , · · · , F n (p n (t))] T : C n → C n , B = (B 1 , · · · , B n ) T ∈ C n , G(p(t − τ 1 (t))) = [G 1 (p 1 (t − τ 1 (t))), · · · , G n (p n (t − τ 1 (t)))] T : C n → C n , D q = C 0 D q t represents the Caputo's fractional derivative, and q ∈ (0, 1), m(t), p(t) represent the state vectors, and the moduli of m i (t) and p i (t) represent respectively the mRNA concentrations and the protein concentrations of the ith node; The parameters a i > 0 and c i > 0 respectively represent the decay rates of mRNA and protein; The moduli of d i and h i mean the translation rates; Both F j (p j (t)) and G j (p j (t − τ 1 (t))) represent the feedback regulation of the protein on the transcription; B j = k∈I j b jk + k∈Ī jb jk , the moduli of b jk andb jk are bounded constants representing the dimensionless transcriptional rate of transcription factor k to j at time t and t − τ 1 (t), and I j ,Ī j , respectively, represent the set of all the k where the transcription factor k is a repressor of gene j at time t and t −τ 1 (t); The matrix W = (w jk ) n×n , E = (e jk ) n×n mean the coupling matrix of the gene network, which are defined as follows: if k is an activator of gene j, The transcriptional delay τ 1 (t) and translational delay τ 2 (t) are bounded continuous functions on R with 0 ≤ τ 1 (t) ≤ τ * 1 and 0 ≤ τ 2 (t) ≤ τ * 2 , here τ * 1 and τ * 2 are positive constants. J 1 (t) and J 2 (t) represent the external input vectors.
Remark 1: Since the above Lemma 3 is not given in [44], for the readers' convenience, we give the proof in the Appendix.

III. MAIN RESULTS
First, we consider the global synchronization of DFGRNs based on the adaptive feedback controller. Let The adaptive feedback controllers are designed as follows: where L kj (k = 1, 2, · · · , 8, j = 1, 2, · · · , n) are arbitrary positive constants,ā kj (t),b kj (t),c kj (t),d kj (t) (k = 1, 2, j = 1, 2, · · · , n) are adaptive coupling strengths. Remark 2: The state feedback control is designed in many dynamic systems such as GRNs. The adaptive feedback control built on adaptive techniques without knowing the values of the unknown parameters in advance is more flexible than the state feedback control, and can adjust the coupling weights adaptively to avoid high feedback gains [11].
It is well known that when t → ∞, the facts , n) mean that the drive system (1) and the response system (2) are synchronized.
From systems (5) and (6), we can get the following error system: From (7) and (8), we have the initial conditions of system (10) are as follows VOLUME 8, 2020 Theorem 1: If assumption 1 holds, then the drive system (1) and the response system (2) are globally asymptotically synchronized based on adaptive feedback controller (9).
So, there exists a constant M > 0 satisfying According to Lemma 3, we have lim t→∞ (t) = 0, that is, the drive system (1) and the response system (2) are globally asymptotically synchronized based on the controller (9). The proof is completed.
Remark 3: Since the well-known Leibniz chain rule is invalid for fractional-order derivative [45], combining Lemma 2, Lyapunov-like function (t), which contains some square terms of the adaptive coupling strengths, become difficult to appear on the right side of D q (t). Therefore, compared with the classical Lyapunov method [45], Lemma 3 is added in the proof of Theorem 1. -valued vectors, W , E, D, H are real-valued matrices, F(p(t)), G(p(t − τ 1 (t))), J 1 (t), J 2 (t), U 1 (t), U 2 (t) are real-valued vector-valued functions and the controller (9) becomes
Remark 6: In Remark 4, if we adopt E = H = 0 in systems (1) and (2), then the controller (17) becomes and assumption 2 becomes: Assumption 3: The functions F j (p j (t)) satisfy the following inequalities where δ 1 j > 0 are Lipschitz constants. Then we have the following conclusion: Corollary 2: If assumption 3 holds, then the drive system (1) without time delay and the response system (2) are globally asymptotically synchronized based on feedback controller (18).
It is well known that when t → ∞, the facts x R j (t) → 0, 1, 2, · · · , n) mean that the drive system (1) and the response system (2) are antisynchronized. From systems (5) and (6), we can get the following error system: Theorem 2: If assumption 1 holds, then the drive system (1) and the response system (2) are globally asymptotically anti-synchronized based on feedback controller (19).
Proof: The proof of Theorem 2 is similar to that of Theorem 1.  matrices, F(p(t)), G(p(t − τ 1 (t))), J 1 (t), J 2 (t), U 1 (t), U 2 (t) are real-valued vector-valued functions and the controller (19) becomes Then we can obtain the following result: Corollary 3: If assumption 2 holds, then the drive system (1) and the response system (2) are globally asymptotically anti-synchronized based on controller (21).
Then we have the following conclusion: Corollary 4: If assumption 3 holds, then the drive system (1) without time delay and the response system (2) are globally asymptotically anti-synchronized based on feedback controller (22).
Remark 10: All the results of our corollaries are still new.

IV. NUMERICAL EXAMPLE
In this section, we give some numerical examples to illustrate the effectiveness of above theoretical results. We take the step-length h = 0.1 for the improved Adams-Bashforth-Moulton predictor-corrector scheme [46], which is available on the fractional-order differential equations with time-varying delays.
According to Theorem 1, the response system (24) will synchronize with the drive system (23) under controller (9). The 2-dimensional synchronization diagrams are shown in Figures 1 -3, where the X -axis represents the time t and the Y -axis represents the real(imaginary) part of the state variables. The phase trajectories are shown in Figures 4 -6, where the X and Y -axes represent the real and imaginary parts of the state variables, respectively. In Figures 1 -6, the drive-response systems (23) and (24) achieve synchronization under controller (9). Figure 7, where the X -axis represents the time t and the Y -axis represents the adaptive gains, shows the adaptive gainsā kj (t),c kj (t),b kj (t) andd kj (t) in controller (9) respectively converge to the corresponding positive constantsâ kj ,ĉ kj ,b kj andd kj (k = 1, 2, j = 1, 2, 3). Also, the synchronization phenomena exists for arbitrary initial conditions and q ∈ (0, 1).
(ii) Use the anti-synchronization controller in the response system (24). In controller (19), we take the same parameters and initial conditions with case (i). According to Theorem 2, the response system (24) will anti-synchronize with the drive system (23) under controller (19). The 2-dimensional anti-synchronization diagrams are shown in Figures 8 -10, where the X -axis represents the time t and the Y -axis represents the real(imaginary) part of the state variables. The phase trajectories are shown in Figures 11 -13, where the X and Y -axes represent the real and imaginary parts of the state variables, respectively. In Figures 8 -13, the drive-response systems (23) and (24) achieve anti-synchronization under controller (19). Figure 14, where the X -axis represents the time t and the Yaxis represents the adaptive gains, shows the adaptive gains a kj (t),c kj (t),b kj (t) andd kj (t) in controller (19)    andd kj (k = 1, 2, j = 1, 2, 3). Also, the anti-synchronization phenomena exists for arbitrary initial conditions and q ∈ (0, 1).
In order to investigate the effects of the fractional-order q on synchronization and anti-synchronization of systems (23) and (24), let error e * (t) = 1 12 |y I i (t)| , T = min{t : e * (t) ≤ε}, whereε is the error limit. We get ''estimated time'' T 1 and T 2 to achieve synchronization and anti-synchronization by calculating T , respectively. For different fractional-order q, the values T 1 and T 2 are seen in Table 1 withε = 0.1, and the trajectories of the errors for synchronization and anti-synchronization are shown in Figure 15. In Figure 15, the X -axis of represents the time t and the Y -axis represents the errors of synchronization or anti-synchronization. From Table 1, we find that the minimum ''estimated time'' T 1 and T 2 increase with the decreasing of the fractional-order q, which means that the fractional-order q can effect the synchronization and anti-synchronization of DFGRNs.     a system that contains 12n fractional-order differential equations. When the time range is [0, 50] and step-length h = 0.1, the numerical result is a 12n×500 matrix. This means that the

V. CONCLUSION
By designing two kinds of adaptive feedback controllers respectively, this paper deals with the global asymptotical synchronization and anti-synchronization for a class of driveresponse system of fractional-order complex-valued GRNs with time-varying delays. And by combining fractionalorder Lyapunov-like function method with the fractionalorder inequality techniques, some sufficient criteria for global asymptotical synchronization and anti-synchronization are derived.
Compared with the other synchronization conditions for GRNs, which expressed via linear matrix inequality (LMI) [17], our sufficient conditions of synchronization and anti-synchronization are simpler and need not to be calculated by the MATLAB LMI toolbox in the simulation part, which avoids the computational complexity caused by high dimension matrix.
In Table 1, we give the minimum ''estimated time'' T 1 and T 2 to achieve synchronization and anti-synchronization, respectively. And we find that the values of T 1 and T 2 increase with the decreasing of the fractional-order q of DFGRNs.
In the future, we will pay more attention to dynamic behaviors of FGRNs with leakage delay, structured uncertainties or stochastic disturbance, such as finite-time synchronization, finite-time stability, Hopf bifurcation, and explore their corresponding practical application.