Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks with mixed delays via state-feedback control

Yongkun Li; Huimei Wang

doi:10.1371/journal.pone.0198297

Abstract

This paper studies the drive-response synchronization for quaternion-valued shunting inhibitory cellular neural networks (QVSICNNs) with mixed delays. First, QVSICNN is decomposed into an equivalent real-valued system in order to avoid the non-commutativity of the multiplicity. Then, the existence of almost periodic solutions is obtained based on the Banach fixed point theorem. An novel state-feedback controller is designed to ensure the global exponential almost periodic synchronization. At the end of the paper, an example is given to illustrate the effectiveness of the obtained results.

Citation: Li Y, Wang H (2018) Almost periodic synchronization of quaternion-valued shunting inhibitory cellular neural networks with mixed delays via state-feedback control. PLoS ONE 13(6): e0198297. https://doi.org/10.1371/journal.pone.0198297

Editor: Takashi Nishikawa, Northwestern University, UNITED STATES

Received: March 19, 2018; Accepted: May 16, 2018; Published: June 7, 2018

Copyright: © 2018 Li, Wang. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: All relevant data are within the paper.

Funding: The funding institution for this work is the National Natural Sciences Foundation of People’s Republic of China. The Grant number is 11361072 and the funder’s website is http://www.nsfc.gov.cn/.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Quaternion was first proposed by Hamilton [1] in 1853. However, because of the non-commutativity of quaternion multiplicity, the development on quaternion was quite slow. Fortunately, with the development of modern science, the quaternion has been widely used in attitude control, quantum mechanics, computer graphics and so on, see [2–5] and references therein. In recent years, quaternion has attracted scholars from many fields, especially, the scholars in the field of neural network research. The quaternion-valued neural networks (QVNNs), as an special case of Clifford-valued neural networks [6], can be thought of as an extension of complex-valued neural networks (CVNNs) and real-valued neural networks (RVNNs). In fact, QVNNs can be applied to engineering and science. A great deal of studies have shown that, for the three dimensional data including color images and body images, via direct coding, QVNNs can do the process with high-efficiency [7]. Indeed, based on the three primary colors and Hamilton rules of quaternion, one can realize the color face recognition efficiently. The quaternion representation treats the color image and dictionary in a holistic manner, while the real representation can only treat the three colors channels separately [8, 9]. Since all of these applications strongly rely on the dynamics of QVNNs, many researchers have studied some dynamical behaviours of QVNNs ([10–15]) recently.

On the one hand, after Bouzerdount and Pinter’s [16] new class of cellular neural networks, namely the shunting inhibitory cellular neural networks (SICNNs), many studies have been focusing on the SICNNs, especially about the dynamical behaviors. Because of the wide applications of SICNNs in psychophysics [17], speech [18], perception [19], robotics [20], adaptive pattern recognition [21, 22], vision [23, 24], and image processing [25], moreover, time delays are unavoidable in a realistic system, there have been extensive results about the sufficient conditions on the problem of the existence and stability of equilibrium, periodic, anti-periodic solutions of SICNNs with time delays, see [26–29] and references therein. Besides, it is well known that the almost periodic phenomenon is more universal than the periodic one in real world. In the past few years, many researchers devoted to study the almost periodic problem of SICNNs with time delays ([30–37]).

On the other hand, synchronization is a very common phenomenon in real systems, which indicates that two or more systems adjust each other to lead to a common dynamical behavior. By synchronization, we can understand an unknown system from the well-known systems. Pecora and Carroll [38] proposed a method to synchronize two identical chaotic systems with different initial values in 1990, from then on, the problem of synchronization has attracted scholars from various fields such as information science [39, 40], secure communication [41, 42] and chemical reactions [43, 44]. In particular, in the field of neural networks, much attention has been focusing on this topic, see [45–54] and references therein. At present, there are some results about the synchronization for complex-valued neural networks [55–58]. However, as far as we know, till now there is still no result about the almost periodic synchronization of SICNNs, not to speak of QVSICNNs.

In this paper, we study the QVSICNNs with time varying and distributed delays.

The paper is organized as follows. In Section 2, some preliminaries and notations are introduced. In Section 3, the sufficient conditions for the existence of almost periodic solutions of system (1) are obtained. In Section 4, the global exponential synchronization is studied. In Section 5, the effectiveness and feasibility of the proposed methods in this paper are shown by a numerical example.

Problem description and preliminaries

We denote the skew field of quaternion by where x^R, x^I, x^J, x^K are real numbers and the elements i, j, k obey the Hamilton’s multiplication rules:

In this paper, the model of the shunting inhibitory cellular neural networks with mixed time delays is defined as follows: (1) where 1 ≤ p ≤ m, 1 ≤ q ≤ n, for the convenience, we denote ; C_pq denotes the cell at the position (p, q) of the lattice; the r-neighborhood of C_pq is defined as and N_s(p, q), N_u(p, q) are similarly specified; is the activity of the cell C_pq, is the external input to C_pq, a_pq(t)>0 represents the passive decay rate of the cell activity; are the connection or coupling strength of postsynaptic activity of the cell transmitted to C_pq, and the activity functions are the continuous functions representing the output or firing rate of the cell C_pq; τ(t) ≥ 0 denotes the transmission time varying delay; K_pq(t) denotes the transmission delay kernels.

The initial conditions associated with system (1) are of the form where are bounded continuous functions.

Now, we introduce some relevant definitions and basic lemmas.

Definition 1. [59] A function is said to be almost periodic if, for any ϵ > 0, it is possible to find a real number l = l(ϵ) > 0, denoting length l(ϵ) of an interval, there exists a number τ = τ(ϵ) in this interval such that |x(t + τ) − x(t)| < ϵ for all .

Denote the set of almost periodic functions by .

Definition 2. A quaternion-valued function is called almost periodic if for every ν ∈ {R, I, J, K}: = Λ, .

Definition 3. [59] Let and A(t) be an n × n matrix function on . Then the linear system (2) is said to admit an exponential dichotomy on if there exist positive constants k_i, α_i, i = 1, 2, projection P, and the fundamental solution matrix X(t) of (2), satisfying where ‖ ⋅ ‖₀ is the matrix norm on .

Let us consider the following almost periodic system (3) where A(t) is an almost periodic matrix function and f(t) is an almost periodic vector function.

Lemma 1. [59] If the linear system (2) admits an exponential dichotomy, then system (3) has a unique almost periodic solution where X(t) is the fundamental solution matrix of (2), I denotes the n × n-identity matrix.

Lemma 2. [59] Let a_p be an almost periodic function on and Then the linear system admits an exponential dichotomy on .

Let , where . Assume the activity functions of (1) can be expressed as where , ν ∈ Λ, and the external input can be expressed as where .

In the following, for a bounded continuous function, we denote , .

In order to overcome the non-commutativity of the quaternion multiplication, according to Hamilton rules, we decompose system (1) into an equivalent real-valued system: (4) (5) (6) (7) where

Denote Applying (4)–(7), we obtain an equivalent real-valued system of the quaternion-valued system (1) as follows: (8) with the initial conditions: where ,

In what follows, we regard (1) as the drive system, and the corresponding response system is expressed as (9) where denotes the state of the response system, is a state-feedback controller, the rest notations are the same as those in system (1) and the initial condition is where are quaternion-valued bounded continuous functions on (−∞, 0].

Denote z_pq(t) = y_pq(t) − x_pq(t), subtracting (1) from (9) yields the following error system: (10) where F(z_kl(t))z_pq(t) = f(y_kl(t))y_pq(t) − f(x_kl(t))x_pq(t), G(z_kl(t − τ(t)))z_pq(t) = g(y_kl(t − τ(t)))y_pq(t) − g(x_kl(t − τ(t)))x_pq(t), H(z_kl(t − u))z_pq(t) = h(y_kl(t − u))y_pq(t) − h(x_kl(t − u))x_pq(t).

In order to show the almost periodic synchronization of the drive-response system, we design the state-feedback controller as follows: where W(z_kl(t − δ(t))) = w(y_kl(t − δ(t)))y_pq(t) − w(x_kl(t − δ(t)))x_pq(t).

Definition 4. The response system (9) and the drive system (1) are said to be globally exponentially synchronized, if there exist positive constants M > 0 and λ > 0 such that where

Analogously, one can decompose (10) into the following real-valued system: (11) (12) (13) (14) where

Remark 1. If is a solution to system (8), then (pq ∈ J) must be a solution to system (1). Thus, the problem of finding an almost periodic solution for (1) is reduced to finding it for system (8). For studying the synchronization of (1) and (9), we just need to consider the exponential stability of system (11)–(14).

Throughout the paper, we assume the following conditions:

(A₁) For , ν ∈ Λ, with M[a_pq] > 0, , , and 1 − α > 0, 1 − β > 0, where .
(A₂) For ν ∈ Λ, f^ν, g^ν, h^ν, and for any , there exist positive constants , , , , such that and
(A₃) For , the delay kernels are continuous and |K_pq(t)|e^λt are integrable on [0,∞) for certain positive constant λ.

Main results

In this section, we establish the sufficient conditions for the existence of almost periodic solutions of system (1), and the sufficient conditions for the global exponential synchronization of the drive system (1) and the response system (9).

Denote , where . For , we define its norm as .

Set . is a Banach space when equipped with the norm .

Theorem 1. Under assumptions (A₁)-(A₃), and

(A₄) there exists a positive constant κ such that where

system (8) has a unique almost periodic solution in .

proof. Given , consider the following linear system (15) Together with (A₁) and Lemma 2, the linear system admits an exponential dichotomy. By Lemma 1, we know that system (15) has a unique almost periodic solution which can be expressed as , where

Now, we define a mapping: with , for .

First, we will show that for any , . Denote For , we have so we have repeat a similar calculation, we obtain Together with the above inequalities, we obtain which following from (A₅) implies that Γ is a self-mapping on .

Next, we show that Γ is a contraction mapping on . For any , we have so Hence, we have Similarly, one can obtain Therefore, which implies that Γ is a contraction mapping. According to the Banach fixed point theorem, Γ has a unique fixed point in , which means that system (8) has a unique almost periodic solution in . The proof is complete.

Theorem 2. Assume (A₁)-(A₄) hold and suppose further that

(A₅) There exists a positive constant λ such that where

Then the drive system (1) and response system (9) are globally exponentially synchronized.

proof. Let us construct a Lyapunov function V(t) as follows where , , ν ∈ Λ and

From (11)–(14), for any t > 0, ν ∈ Λ, , we have

Computing the derivative of V₁(t) along the solution of the error system (10), we obtain (16)

Repeat a similar calculation, we obtain (17) It follows from (A₅), (16) and (17) that which implies that V(t) ≤ V(0) for all t ≥ 0.

On the other hand, we have We also have where Therefore, the drive system (1) and the response system (9) are globally exponentially synchronized. The proof is complete.

A numerical example

In this section, an example is shown for the effectiveness of the proposed method in this paper.

Example 1. If the following QVSICNN as the drive system: (18) and the corresponding response system is defined as (19) where K_pq(u) = (cos u)e^−2u, p, q = 1, 2, r = s = u = v = 1, , , and the coefficients are as follows: We have Thus, (A₁)-(A₃) hold. Setting κ = 2, for p, q = 1, 2, we have which implies that (A₄) is satisfied. Therefore, the drive system (18) has a unique almost periodic solution. Moreover, take λ = 1, we have Thus, (A₅) is also satisfied. Therefore, (18) and (19) are globally exponentially synchronized (see Figs 1–4).

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Fig 1. Transient states of four parts of x_pq, p, q = 1, 2.

https://doi.org/10.1371/journal.pone.0198297.g001

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Fig 2. Transient states of four parts of y_pq, p, q = 1, 2.

https://doi.org/10.1371/journal.pone.0198297.g002

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Fig 3. Synchronization.

https://doi.org/10.1371/journal.pone.0198297.g003

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Fig 4. Curves of

(p, q = 1, 2, ν ∈ Λ) in 3-dimensional space for synchronization case.

https://doi.org/10.1371/journal.pone.0198297.g004

Conclusion

In this paper, a class of QVSICNNs with mixed delays is studied. To the best of our knowledge, this is the first on studying the problem. Since QVSICNNs include RVSICNNs and CVSICNNs as special cases, our method of this paper can be applied to study the almost periodic synchronization problem of other types of neural networks including RVNNs and CVNNs.

In this paper, the almost periodic synchronization of a class of QVSICNNs with mixed delays is studied. To the best of our knowledge, this is the first on studying the problem. Since QVSICNNs include RVSICNNs and CVSICNNs as special cases, our method of this paper can be applied to study the almost periodic synchronization problem of other types of neural networks including RVNNs and CVNNs.

Acknowledgments

This work is supported by the National Natural Sciences Foundation of People’s Republic of China under Grant 11361072.

References

1. Sudbery A. Quaternionic analysis. Mathematical Proceedings of the Cambridge Philosophical Society. 1979;85(2):199–225.
- View Article
- Google Scholar
2. Adler SL, Finkelstein DR. Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press; 1995.
3. Su BC, Faraway JJ. Modeling head and hand orientation during motion using quaternions. Sae Transactions. 2004;.
4. Chou JCK. Quaternion kinematic and dynamic differential equations. IEEE Transactions on Robotics and Automation. 1992;8(1):53–64.
- View Article
- Google Scholar
5. Mukundan R. Quaternions: From classical mechanics to computer graphics, and beyond. In: Proceedings of the 7th Asian Technology Conference in Mathematics; 2002.
6. Liu Y, Xu P, Lu J, Liang J. Global stability of Clifford-valued recurrent neural networks with time delays. Nonlinear Dynamics. 2016;84(2):767–777.
- View Article
- Google Scholar
7. Hiromi K, Teijiro I, Nobuyuki M, Yuzo O, Kazuaki M. A new scheme for color night vision by quaternion neural network. In: Proceedings of the 2nd International Conference on Autonomous Robots and Agents; 2004. p. 101–106.
8. Zou C, Kou KI, Wang Y. Quaternion collaborative and sparse representation with application to color face recognition. IEEE Transactions on Image Processing. 2016;25(7):3287–3302.
- View Article
- Google Scholar
9. Chen B, Shu H, Coatrieux G, Chen G, Sun X, Coatrieux JL. Color image analysis by quaternion-type moments. Journal of Mathematical Imaging and Vision. 2015;51(1):124–144.
- View Article
- Google Scholar
10. Liu Y, Zhang D, Lu J, Cao J. Global μ-stability criteria for quaternion-valued neural networks with unbounded time-varying delays. Information Sciences. 2016;360:273–288.
- View Article
- Google Scholar
11. Liu Y, Zhang D, Lu J. Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays. Nonlinear Dynamics. 2016; 87(1):553–565.
- View Article
- Google Scholar
12. Zhang D, Kou KI, Liu Y, Cao J. Decomposition approach to the stability of recurrent neural networks with asynchronous time delays in quaternion field. Neural Networks. 2017;94:55. pmid:28753445
- View Article
- PubMed/NCBI
- Google Scholar
13. Li Y, Meng X. Existence and global exponential stability of pseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on time scales. Complexity. 2017;2017(10):1–15.
- View Article
- Google Scholar
14. Li Y, Qin J. Existence and global exponential stability of periodic solutions for quaternion-valued cellular neural networks with time-varying delays. Neurocomputing. 2018;292:91–103.
- View Article
- Google Scholar
15. Popa CA, Kaslik E. Multistability and multiperiodicity in impulsive hybrid quaternion-valued neural networks with mixed delays. Neural Networks. 2018;99:1–18. pmid:29306800
- View Article
- PubMed/NCBI
- Google Scholar
16. Bouzerdoum A, Pinter RB. Shunting inhibitory cellular neural networks: derivation and stability analysis. IEEE Transactions on Circuits and Systems. 1993;40(3):215–221.
- View Article
- Google Scholar
17. Pinter RB. The electrophysiological bases for linear and for nonlinear product term lateral inhibition and the consequences for wide field textured stimuli. Journal of Theoretical Biology. 1983;105(2):233–243. pmid:6656281
- View Article
- PubMed/NCBI
- Google Scholar
18. Grossberg S. Neural Networks and Natural Intelligence. MIT Press; 1992.
19. Bouzerdoum A, Pinter RB. A shunting inhibitory motion detector that can account for the functional characteristics of fly motion-sensitive interneurons. In: IJCNN International Joint Conference on Neural Networks; 1990. p. 149–153.
20. Gaudiano P, Grossberg S. Vector associative maps: Unsupervised real-time error-based learning and control of movement trajectories. Neural Networks. 1991;4(2):147–183.
- View Article
- Google Scholar
21. Carpenter GA, Grossberg S. The ART of adaptive pattern recognition by a self-organizing neural network. Computer. 1988;21(3):77–88.
- View Article
- Google Scholar
22. Pinter RB. Adaptation of receptive field spatial organization via multiplicative lateral inhibition. Journal of Theoretical Biology. 1984;110(3):435–444. pmid:6503309
- View Article
- PubMed/NCBI
- Google Scholar
23. Pinter RB. Product term nonlinear lateral inhibition enhances visual selectivity for small objects or edges. Journal of Theoretical Biology. 1983;100(3):525–531. pmid:6834868
- View Article
- PubMed/NCBI
- Google Scholar
24. Bouzerdoum A. Nonlinear lateral inhibition applied to motion detection in the fly visual system. Nonlinear vision. 1992; p. 423–450.
- View Article
- Google Scholar
25. Cheung HN, Bouzerdoum A, Newland W. Properties of shunting inhibitory cellular neural networks for colour image enhancement. In: Proceedings of 6th International Conference on Neural Information Processing. Perth, Western Australia: IEEE; 1999. p. 1219–1223.
26. Jiang A. Exponential convergence for shunting inhibitory cellular neural networks with oscillating coefficients in leakage terms. Neurocomputing. 2015;165:159–162.
- View Article
- Google Scholar
27. Aouiti C. Neutral impulsive shunting inhibitory cellular neural networks with time-varying coefficients and leakage delays. Cognitive neurodynamics. 2016;10(6):573–591. pmid:27891204
- View Article
- PubMed/NCBI
- Google Scholar
28. Li Y, Liu C, Zhu L. Global exponential stability of periodic solution for shunting inhibitory CNNs with delays. Physics Letters A. 2005;337(1-2):46–54.
- View Article
- Google Scholar
29. Li Y, Shu J. Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales. Communications in Nonlinear Science and Numerical Simulation. 2011;16(8):3326–3336.
- View Article
- Google Scholar
30. Huang X, Cao J. Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delay. Physics Letters A. 2003;314(3):222–231.
- View Article
- Google Scholar
31. Liu Y, You Z, Cao L. Almost periodic solution of shunting inhibitory cellular neural networks with time varying and continuously distributed delays. Physics Letters A. 2007;364(1):17–28.
- View Article
- Google Scholar
32. Li Y, Wang C. Almost periodic solutions of shunting inhibitory cellular neural networks on time scales. Communications in Nonlinear Science and Numerical Simulation. 2012;17(8):3258–3266.
- View Article
- Google Scholar
33. Wang W, Liu B. Global exponential stability of pseudo almost periodic solutions for SICNNs with time-varying leakage delays. Abstract and Applied Analysis. 2014;2014(31):1–17.
- View Article
- Google Scholar
34. M’Hamdi MS, Aouiti C, Touiti A, Alimi AM, Sansel V. Weighted pseudo almost-periodic solutions of shunting inhibitory cellular neural networks with mixed delays. Acta Mathematica Scientia. 2016;36(6):1662–1682.
- View Article
- Google Scholar
35. Wang P, Li B, Li Y. Square-mean almost periodic solutions for impulsive stochastic shunting inhibitory cellular neural networks with delays. Neurocomputing. 2015;167:76–82.
- View Article
- Google Scholar
36. Xu C, Li P. On anti-periodic solutions for neutral shunting inhibitory cellular neural networks with time-varying delays and D operator. Neurocomputing. 2018;275:377–382.
- View Article
- Google Scholar
37. Zhang A. Pseudo almost periodic solutions for SICNNs with oscillating leakage coefficients and complex deviating arguments. Neural Processing Letters. 2017;45(1):183–196.
- View Article
- Google Scholar
38. Pecora LM, Calloll TL. Synchronization in chaotic systems. Physical Review Letters. 1990;64(8):821–824. pmid:10042089
- View Article
- PubMed/NCBI
- Google Scholar
39. Hong H. Periodic synchronization and chimera in conformist and contrarian oscillators. Physical Review E Statistical Nonlinear and Soft Matter Physics. 2014;89(6):1–37.
- View Article
- Google Scholar
40. Ling L, Li C, Chen L, Wei L. Lag projective synchronization of a class of complex network constituted nodes with chaotic behavior. Communications in Nonlinear Science and Numerical Simulation. 2014;19(8):2843–2849.
- View Article
- Google Scholar
41. Yang T, Chua LO. Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication. IEEE Transactions on Circuits and Systems. 1997;44(10):976–988.
- View Article
- Google Scholar
42. Lu J, Wu X, Lü J. Synchronization of a unified chaotic system and the application in secure communication. Physics Letters A. 2002;305(6):365–370.
- View Article
- Google Scholar
43. Vaidyanathan S. Global chaos synchronization of chemical chaotic reactors via novel sliding mode control method. International Journal of Chemtech Research. 2015;8(7):209–221.
- View Article
- Google Scholar
44. Vaidyanathan S. Integral sliding mode control design for the gGlobal chaos synchronization of identical novel chemical chaotic reactor systems. International Journal of Chemtech Research. 2015;8(11):684–699.
- View Article
- Google Scholar
45. Michalak A, Nowakowski A. Finite-time stability and finite-time synchronization of neural network-Dual approach. Journal of the Franklin Institute. 2017;354(18).
- View Article
- Google Scholar
46. Li R, Cao J, Alsaedi A, Alsaadi F. Exponential and fixed-time synchronization of Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms. Applied Mathematics and Computation. 2017;313:37–51.
- View Article
- Google Scholar
47. Luo Y, Shu L, Zhou B. Global exponential synchronization of nonlinearly coupled complex dynamical networks with time-varying coupling delays. Complexity. 2017;2017:1–10.
- View Article
- Google Scholar
48. Zhang J, Gao Y. Synchronization of coupled neural networks with time-varying delay. Neurocomputing. 2017;219:154–162.
- View Article
- Google Scholar
49. Hu C, Yu J, Jiang H, Teng Z. Exponential stabilization and synchronization of neural networks with time-varying delays via periodically intermittent control. Nonlinearity. 2010;23(10):2369–2391.
- View Article
- Google Scholar
50. Cai Z, Huang L, Wang D, Zhang L. Periodic synchronization in delayed memristive neural networks based on Filippov systems. Journal of the Franklin Institute. 2015;352(10):4638–4663.
- View Article
- Google Scholar
51. Wu W, Chen T. Global synchronization criteria of linearly coupled neural network systems with time-varying coupling. IEEE Transactions on Neural Networks. 2008;19(2):319–332. pmid:18269962
- View Article
- PubMed/NCBI
- Google Scholar
52. Ma Q, Xu S, Zou Y. Stability and synchronization for Markovian jump neural networks with partly unknown transition probabilities. Neurocomputing. 2011;74:3404–3411.
- View Article
- Google Scholar
53. Ozcan N, Ali MS, Yogambigai J, Zhu Q, Arik S. Robust synchronization of uncertain Markovian jump complex dynamical networks with time-varying delays and reaction-diffusion terms via sampled-data control. Journal of the Franklin Institute. 2018;355:1192–1216.
- View Article
- Google Scholar
54. Ali MS, Yogambigai J. Passivity-based synchronization of stochastic switched complex dynamical networks with additive time-varying delays via impulsive control. Neurocomputing. 2017;273:209–221.
- View Article
- Google Scholar
55. Ding X, Cao J, Alsaedi A, Alsaadi FE, Hayat T. Robust fixed-time synchronization for uncertain complex-valued neural networks with discontinuous activation functions. Neural Networks. 2017;90:42. pmid:28388472
- View Article
- PubMed/NCBI
- Google Scholar
56. Yang X, Li C, Huang T, Song Q, Huang J. Synchronization of fractional-order memristor-based complex-valued neural networks with uncertain parameters and time delays. Chaos, Solitons and Fractals. 2018;110:105–123.
- View Article
- Google Scholar
57. Zhou C, Zhang W, Yang X, Xu C, Feng J. Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations. Neural Processing Letters. 2017;46(1):271–291.
- View Article
- Google Scholar
58. Zhang L, Yang X, Xu C, Feng J. Exponential synchronization of complex-valued complex networks with time-varying delays and stochastic perturbations via time-delayed impulsive control. Applied Mathematics and Computation. 2017;306:22–30.
- View Article
- Google Scholar
59. Fink AM. Almost Periodic Differential Equations. Berlin, Germany: Springer-Verlag; 1974.

[ref1] 1. Sudbery A. Quaternionic analysis. Mathematical Proceedings of the Cambridge Philosophical Society. 1979;85(2):199–225.
View Article
Google Scholar

[2] View Article

[3] Google Scholar

[ref2] 2. Adler SL, Finkelstein DR. Quaternionic Quantum Mechanics and Quantum Fields. Oxford University Press; 1995.

[ref3] 3. Su BC, Faraway JJ. Modeling head and hand orientation during motion using quaternions. Sae Transactions. 2004;.

[ref4] 4. Chou JCK. Quaternion kinematic and dynamic differential equations. IEEE Transactions on Robotics and Automation. 1992;8(1):53–64.
View Article
Google Scholar

[7] View Article

[8] Google Scholar

[ref5] 5. Mukundan R. Quaternions: From classical mechanics to computer graphics, and beyond. In: Proceedings of the 7th Asian Technology Conference in Mathematics; 2002.

[ref6] 6. Liu Y, Xu P, Lu J, Liang J. Global stability of Clifford-valued recurrent neural networks with time delays. Nonlinear Dynamics. 2016;84(2):767–777.
View Article
Google Scholar

[11] View Article

[12] Google Scholar

[ref7] 7. Hiromi K, Teijiro I, Nobuyuki M, Yuzo O, Kazuaki M. A new scheme for color night vision by quaternion neural network. In: Proceedings of the 2nd International Conference on Autonomous Robots and Agents; 2004. p. 101–106.

[ref8] 8. Zou C, Kou KI, Wang Y. Quaternion collaborative and sparse representation with application to color face recognition. IEEE Transactions on Image Processing. 2016;25(7):3287–3302.
View Article
Google Scholar

[15] View Article

[16] Google Scholar

[ref9] 9. Chen B, Shu H, Coatrieux G, Chen G, Sun X, Coatrieux JL. Color image analysis by quaternion-type moments. Journal of Mathematical Imaging and Vision. 2015;51(1):124–144.
View Article
Google Scholar

[18] View Article

[19] Google Scholar

[ref10] 10. Liu Y, Zhang D, Lu J, Cao J. Global μ-stability criteria for quaternion-valued neural networks with unbounded time-varying delays. Information Sciences. 2016;360:273–288.
View Article
Google Scholar

[21] View Article

[22] Google Scholar

[ref11] 11. Liu Y, Zhang D, Lu J. Global exponential stability for quaternion-valued recurrent neural networks with time-varying delays. Nonlinear Dynamics. 2016; 87(1):553–565.
View Article
Google Scholar

[24] View Article

[25] Google Scholar

[ref12] 12. Zhang D, Kou KI, Liu Y, Cao J. Decomposition approach to the stability of recurrent neural networks with asynchronous time delays in quaternion field. Neural Networks. 2017;94:55. pmid:28753445
View Article
PubMed/NCBI
Google Scholar

[27] View Article

[28] PubMed/NCBI

[29] Google Scholar

[ref13] 13. Li Y, Meng X. Existence and global exponential stability of pseudo almost periodic solutions for neutral type quaternion-valued neural networks with delays in the leakage term on time scales. Complexity. 2017;2017(10):1–15.
View Article
Google Scholar

[31] View Article

[32] Google Scholar

[ref14] 14. Li Y, Qin J. Existence and global exponential stability of periodic solutions for quaternion-valued cellular neural networks with time-varying delays. Neurocomputing. 2018;292:91–103.
View Article
Google Scholar

[34] View Article

[35] Google Scholar

[ref15] 15. Popa CA, Kaslik E. Multistability and multiperiodicity in impulsive hybrid quaternion-valued neural networks with mixed delays. Neural Networks. 2018;99:1–18. pmid:29306800
View Article
PubMed/NCBI
Google Scholar

[37] View Article

[38] PubMed/NCBI

[39] Google Scholar

[ref16] 16. Bouzerdoum A, Pinter RB. Shunting inhibitory cellular neural networks: derivation and stability analysis. IEEE Transactions on Circuits and Systems. 1993;40(3):215–221.
View Article
Google Scholar

[41] View Article

[42] Google Scholar

[ref17] 17. Pinter RB. The electrophysiological bases for linear and for nonlinear product term lateral inhibition and the consequences for wide field textured stimuli. Journal of Theoretical Biology. 1983;105(2):233–243. pmid:6656281
View Article
PubMed/NCBI
Google Scholar

[44] View Article

[45] PubMed/NCBI

[46] Google Scholar

[ref18] 18. Grossberg S. Neural Networks and Natural Intelligence. MIT Press; 1992.

[ref19] 19. Bouzerdoum A, Pinter RB. A shunting inhibitory motion detector that can account for the functional characteristics of fly motion-sensitive interneurons. In: IJCNN International Joint Conference on Neural Networks; 1990. p. 149–153.

[ref20] 20. Gaudiano P, Grossberg S. Vector associative maps: Unsupervised real-time error-based learning and control of movement trajectories. Neural Networks. 1991;4(2):147–183.
View Article
Google Scholar

[50] View Article

[51] Google Scholar

[ref21] 21. Carpenter GA, Grossberg S. The ART of adaptive pattern recognition by a self-organizing neural network. Computer. 1988;21(3):77–88.
View Article
Google Scholar

[53] View Article

[54] Google Scholar

[ref22] 22. Pinter RB. Adaptation of receptive field spatial organization via multiplicative lateral inhibition. Journal of Theoretical Biology. 1984;110(3):435–444. pmid:6503309
View Article
PubMed/NCBI
Google Scholar

[56] View Article

[57] PubMed/NCBI

[58] Google Scholar

[ref23] 23. Pinter RB. Product term nonlinear lateral inhibition enhances visual selectivity for small objects or edges. Journal of Theoretical Biology. 1983;100(3):525–531. pmid:6834868
View Article
PubMed/NCBI
Google Scholar

[60] View Article

[61] PubMed/NCBI

[62] Google Scholar

[ref24] 24. Bouzerdoum A. Nonlinear lateral inhibition applied to motion detection in the fly visual system. Nonlinear vision. 1992; p. 423–450.
View Article
Google Scholar

[64] View Article

[65] Google Scholar

[ref25] 25. Cheung HN, Bouzerdoum A, Newland W. Properties of shunting inhibitory cellular neural networks for colour image enhancement. In: Proceedings of 6th International Conference on Neural Information Processing. Perth, Western Australia: IEEE; 1999. p. 1219–1223.

[ref26] 26. Jiang A. Exponential convergence for shunting inhibitory cellular neural networks with oscillating coefficients in leakage terms. Neurocomputing. 2015;165:159–162.
View Article
Google Scholar

[68] View Article

[69] Google Scholar

[ref27] 27. Aouiti C. Neutral impulsive shunting inhibitory cellular neural networks with time-varying coefficients and leakage delays. Cognitive neurodynamics. 2016;10(6):573–591. pmid:27891204
View Article
PubMed/NCBI
Google Scholar

[71] View Article

[72] PubMed/NCBI

[73] Google Scholar

[ref28] 28. Li Y, Liu C, Zhu L. Global exponential stability of periodic solution for shunting inhibitory CNNs with delays. Physics Letters A. 2005;337(1-2):46–54.
View Article
Google Scholar

[75] View Article

[76] Google Scholar

[ref29] 29. Li Y, Shu J. Anti-periodic solutions to impulsive shunting inhibitory cellular neural networks with distributed delays on time scales. Communications in Nonlinear Science and Numerical Simulation. 2011;16(8):3326–3336.
View Article
Google Scholar

[78] View Article

[79] Google Scholar

[ref30] 30. Huang X, Cao J. Almost periodic solution of shunting inhibitory cellular neural networks with time-varying delay. Physics Letters A. 2003;314(3):222–231.
View Article
Google Scholar

[81] View Article

[82] Google Scholar

[ref31] 31. Liu Y, You Z, Cao L. Almost periodic solution of shunting inhibitory cellular neural networks with time varying and continuously distributed delays. Physics Letters A. 2007;364(1):17–28.
View Article
Google Scholar

[84] View Article

[85] Google Scholar

[ref32] 32. Li Y, Wang C. Almost periodic solutions of shunting inhibitory cellular neural networks on time scales. Communications in Nonlinear Science and Numerical Simulation. 2012;17(8):3258–3266.
View Article
Google Scholar

[87] View Article

[88] Google Scholar

[ref33] 33. Wang W, Liu B. Global exponential stability of pseudo almost periodic solutions for SICNNs with time-varying leakage delays. Abstract and Applied Analysis. 2014;2014(31):1–17.
View Article
Google Scholar

[90] View Article

[91] Google Scholar

[ref34] 34. M’Hamdi MS, Aouiti C, Touiti A, Alimi AM, Sansel V. Weighted pseudo almost-periodic solutions of shunting inhibitory cellular neural networks with mixed delays. Acta Mathematica Scientia. 2016;36(6):1662–1682.
View Article
Google Scholar

[93] View Article

[94] Google Scholar

[ref35] 35. Wang P, Li B, Li Y. Square-mean almost periodic solutions for impulsive stochastic shunting inhibitory cellular neural networks with delays. Neurocomputing. 2015;167:76–82.
View Article
Google Scholar

[96] View Article

[97] Google Scholar

[ref36] 36. Xu C, Li P. On anti-periodic solutions for neutral shunting inhibitory cellular neural networks with time-varying delays and D operator. Neurocomputing. 2018;275:377–382.
View Article
Google Scholar

[99] View Article

[100] Google Scholar

[ref37] 37. Zhang A. Pseudo almost periodic solutions for SICNNs with oscillating leakage coefficients and complex deviating arguments. Neural Processing Letters. 2017;45(1):183–196.
View Article
Google Scholar

[102] View Article

[103] Google Scholar

[ref38] 38. Pecora LM, Calloll TL. Synchronization in chaotic systems. Physical Review Letters. 1990;64(8):821–824. pmid:10042089
View Article
PubMed/NCBI
Google Scholar

[105] View Article

[106] PubMed/NCBI

[107] Google Scholar

[ref39] 39. Hong H. Periodic synchronization and chimera in conformist and contrarian oscillators. Physical Review E Statistical Nonlinear and Soft Matter Physics. 2014;89(6):1–37.
View Article
Google Scholar

[109] View Article

[110] Google Scholar

[ref40] 40. Ling L, Li C, Chen L, Wei L. Lag projective synchronization of a class of complex network constituted nodes with chaotic behavior. Communications in Nonlinear Science and Numerical Simulation. 2014;19(8):2843–2849.
View Article
Google Scholar

[112] View Article

[113] Google Scholar

[ref41] 41. Yang T, Chua LO. Impulsive stabilization for control and synchronization of chaotic systems: theory and application to secure communication. IEEE Transactions on Circuits and Systems. 1997;44(10):976–988.
View Article
Google Scholar

[115] View Article

[116] Google Scholar

[ref42] 42. Lu J, Wu X, Lü J. Synchronization of a unified chaotic system and the application in secure communication. Physics Letters A. 2002;305(6):365–370.
View Article
Google Scholar

[118] View Article

[119] Google Scholar

[ref43] 43. Vaidyanathan S. Global chaos synchronization of chemical chaotic reactors via novel sliding mode control method. International Journal of Chemtech Research. 2015;8(7):209–221.
View Article
Google Scholar

[121] View Article

[122] Google Scholar

[ref44] 44. Vaidyanathan S. Integral sliding mode control design for the gGlobal chaos synchronization of identical novel chemical chaotic reactor systems. International Journal of Chemtech Research. 2015;8(11):684–699.
View Article
Google Scholar

[124] View Article

[125] Google Scholar

[ref45] 45. Michalak A, Nowakowski A. Finite-time stability and finite-time synchronization of neural network-Dual approach. Journal of the Franklin Institute. 2017;354(18).
View Article
Google Scholar

[127] View Article

[128] Google Scholar

[ref46] 46. Li R, Cao J, Alsaedi A, Alsaadi F. Exponential and fixed-time synchronization of Cohen-Grossberg neural networks with time-varying delays and reaction-diffusion terms. Applied Mathematics and Computation. 2017;313:37–51.
View Article
Google Scholar

[130] View Article

[131] Google Scholar

[ref47] 47. Luo Y, Shu L, Zhou B. Global exponential synchronization of nonlinearly coupled complex dynamical networks with time-varying coupling delays. Complexity. 2017;2017:1–10.
View Article
Google Scholar

[133] View Article

[134] Google Scholar

[ref48] 48. Zhang J, Gao Y. Synchronization of coupled neural networks with time-varying delay. Neurocomputing. 2017;219:154–162.
View Article
Google Scholar

[136] View Article

[137] Google Scholar

[ref49] 49. Hu C, Yu J, Jiang H, Teng Z. Exponential stabilization and synchronization of neural networks with time-varying delays via periodically intermittent control. Nonlinearity. 2010;23(10):2369–2391.
View Article
Google Scholar

[139] View Article

[140] Google Scholar

[ref50] 50. Cai Z, Huang L, Wang D, Zhang L. Periodic synchronization in delayed memristive neural networks based on Filippov systems. Journal of the Franklin Institute. 2015;352(10):4638–4663.
View Article
Google Scholar

[142] View Article

[143] Google Scholar

[ref51] 51. Wu W, Chen T. Global synchronization criteria of linearly coupled neural network systems with time-varying coupling. IEEE Transactions on Neural Networks. 2008;19(2):319–332. pmid:18269962
View Article
PubMed/NCBI
Google Scholar

[145] View Article

[146] PubMed/NCBI

[147] Google Scholar

[ref52] 52. Ma Q, Xu S, Zou Y. Stability and synchronization for Markovian jump neural networks with partly unknown transition probabilities. Neurocomputing. 2011;74:3404–3411.
View Article
Google Scholar

[149] View Article

[150] Google Scholar

[ref53] 53. Ozcan N, Ali MS, Yogambigai J, Zhu Q, Arik S. Robust synchronization of uncertain Markovian jump complex dynamical networks with time-varying delays and reaction-diffusion terms via sampled-data control. Journal of the Franklin Institute. 2018;355:1192–1216.
View Article
Google Scholar

[152] View Article

[153] Google Scholar

[ref54] 54. Ali MS, Yogambigai J. Passivity-based synchronization of stochastic switched complex dynamical networks with additive time-varying delays via impulsive control. Neurocomputing. 2017;273:209–221.
View Article
Google Scholar

[155] View Article

[156] Google Scholar

[ref55] 55. Ding X, Cao J, Alsaedi A, Alsaadi FE, Hayat T. Robust fixed-time synchronization for uncertain complex-valued neural networks with discontinuous activation functions. Neural Networks. 2017;90:42. pmid:28388472
View Article
PubMed/NCBI
Google Scholar

[158] View Article

[159] PubMed/NCBI

[160] Google Scholar

[ref56] 56. Yang X, Li C, Huang T, Song Q, Huang J. Synchronization of fractional-order memristor-based complex-valued neural networks with uncertain parameters and time delays. Chaos, Solitons and Fractals. 2018;110:105–123.
View Article
Google Scholar

[162] View Article

[163] Google Scholar

[ref57] 57. Zhou C, Zhang W, Yang X, Xu C, Feng J. Finite-time synchronization of complex-valued neural networks with mixed delays and uncertain perturbations. Neural Processing Letters. 2017;46(1):271–291.
View Article
Google Scholar

[165] View Article

[166] Google Scholar

[ref58] 58. Zhang L, Yang X, Xu C, Feng J. Exponential synchronization of complex-valued complex networks with time-varying delays and stochastic perturbations via time-delayed impulsive control. Applied Mathematics and Computation. 2017;306:22–30.
View Article
Google Scholar

[168] View Article

[169] Google Scholar

[ref59] 59. Fink AM. Almost Periodic Differential Equations. Berlin, Germany: Springer-Verlag; 1974.

Figures

Abstract

Introduction

Problem description and preliminaries

Main results

A numerical example

Conclusion

Acknowledgments

References