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Mean-Square Exponential Input-to-State Stability of Stochastic Gene Regulatory Networks with Multiple Time Delays

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Abstract

This paper is concerned with the input-to-state stability of stochastic gene regulatory networks with multiple time delays. It is well acknowledged that stochastic systems can accurately describe some complex systems with random disturbances. So it is significant that stochastic systems are applied to model gene regulatory networks because of the complex relationship between genes and proteins from a micro perspective. Considering the differences between stochastic differential equations and ordinary differential equations, we introduce the new stability criterion which is different from the general stability criteria. Making use of Lyapunov functionals, It\(\hat{o}\) formula and Dynkin formula, we present sufficient conditions to guarantee that the proposed system is mean-square exponentially input-to-state stable. Moreover, numerical examples are given to illustrate validity and feasibility of the obtained results.

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References

  1. Achimescu S, Lipan O (2006) Signal propagation in nonlinear stochastic gene regulatory networks. Syst Biol 153(3):120–134

    Google Scholar 

  2. Arda HE, Benitez CM, Kim SK (2013) Gene regulatory networks governing pancreas development. Dev Cell 25(1):5–13

    Google Scholar 

  3. Buckingham M, Rigby PW (2014) Gene regulatory networks and transcriptional mechanisms that control myogenesis. Dev Cell 28(3):225–238

    Google Scholar 

  4. Chen J, Ahn CK, Karimi HR, Cao JD, Qi WH (2018) An event-based asynchronous approach to Markov jump systems with hidden mode detections and missing measurements. IEEE Trans Syst Man Cybern Syst. https://doi.org/10.1109/TSMC.2018.2866906

    Article  Google Scholar 

  5. Cheng J, Park JH, Cao JD, Zhang D (2018) Quantized \(\cal{H}_{\infty }\) filtering for switched linear parameter-varying systems with sojourn probabilities and unreliable communication channels. Inf Sci 446:289–302

    MathSciNet  Google Scholar 

  6. Erban R, Frewen TA, Wang X, Elston TC, Coifman R, Nadler B, Kevrekidis IG (2007) Variable-free exploration of stochastic models: A gene regulatory network example. J Chem Phys 126(15):155103–155115

    Google Scholar 

  7. Fournier T, Gabriel JP, Pasquier J (2009) Stochastic models and numerical algorithms for a class of regulatory gene networks. Bull Math Biol 71(6):1394–1431

    MathSciNet  MATH  Google Scholar 

  8. Gebert C, Radde N, Weber GW (2005) Modeling gene regulatory networks with piecewise linear differential equations. Eur J Oper Res 181(3):1148–1165

    MathSciNet  MATH  Google Scholar 

  9. Isaac C, Thanner MP, Wiktor J, Antonio DS (2013) Detecting cellular reprogramming determinants by differential stability analysis of gene regulatory networks. BMC Syst Biol 7(1):140–152

    Google Scholar 

  10. Kim TH, Hori Y, Hara S (2011) Robust stability analysis of genecprotein regulatory networks with cyclic activationcrepression interconnections. Syst Control Lett 60(6):373–382

    MATH  Google Scholar 

  11. Knight JM, Datta A, Dougherty ER (2012) Generating stochastic gene regulatory networks consistent with pathway information and steady-state behavior. Neural Process Lett 59(6):1701–1710

    Google Scholar 

  12. Kutumova EO (2012) The cycles of functioning of a discrete model of a regulatory circuit of a gene network with threshold functions. J Appl Ind Math 6(2):203–209

    MathSciNet  Google Scholar 

  13. Lai Q (2018) Stability and bifurcation of delayed bidirectional gene regulatory networks with negative feedback loops. Chin J Phys 56(3):1064–1073

    MathSciNet  Google Scholar 

  14. Lakshmanan S, Rihan FA, Rakkiyappan R, Park JH (2014) Stability analysis of the differential genetic regulatory networks model with time-varying delays and Markovian jumping parameters. Nonlinear Anal Hybrid Syst 14:1–15

    MathSciNet  MATH  Google Scholar 

  15. Lee TH, Park JH (2017) Improved criteria for sampled-data synchronization of chaotic Lur’e systems using two new approachess. Nonlinear Anal Hybrid Syst 24:132–145

    MathSciNet  MATH  Google Scholar 

  16. Lee TH, Park JH, Kwon OM (2013a) Stochastic sampled-data control for state estimation of time-varying delayed neural networks. Neural Netw 46:99–108

    MATH  Google Scholar 

  17. Lee TH, Park JH, Lee SM, Kwon OM (2013b) Robust synchronisation of chaotic systems with randomly occurring uncertainties via stochastic sampled-data controls. Int J Control 86(1):107–119

    MathSciNet  MATH  Google Scholar 

  18. Li CG, Chen LN, Aihara K (2006) Stability of genetic networks with SUM regulatory logic: Lur’e system and LMI approach. Neural Process Lett 53(11):2451–2458

    MATH  Google Scholar 

  19. Liang J, Lam J, Wang Z (2009) State estimation for Markov-type genetic regulatory networks with delays and uncertain mode transition rates. Phys Lett A 373(47):4328–4337

    MathSciNet  MATH  Google Scholar 

  20. Luo Q, Gong YY (2017) Stability of gene regulatory networks with Levy noise. Sci China Inf Sci 60(7):1–13

    Google Scholar 

  21. Menolascina F, Bernardo MD, Bernardo DD (2018) Analysis, design and implementation of a novel scheme for in-vivo control of synthetic gene regulatory networks. Automatica 47(6):1265–1270

    MathSciNet  MATH  Google Scholar 

  22. Njah H, Jamoussi S (2015) Weighted ensemble learning of Bayesian network for gene regulatory networks. Neurocomputing 150(20):404–416

    Google Scholar 

  23. Ohara T, Hearn TJ, Webb AA, Satake A (2018) Gene regulatory network models in response to sugars in the plant circadian system. J Theor Biol 457(14):137–151

    MathSciNet  MATH  Google Scholar 

  24. Pan W, Wang ZD, Gao HJ, Li YR, Du M (2010) On multistability of delayed genetic regulatory networks with multivariable regulation functions. Math Biosci 228(1):100–109

    MathSciNet  MATH  Google Scholar 

  25. Pinho R, Garcia V, Irimia M, Feldman MW (2014) Stability depends on positive autoregulation in Boolean gene regulatory networks. PLoS Comput Biol 10(11):572–587

    Google Scholar 

  26. Ren F, Cao J (2008a) Asymptotic and robust stability of genetic regulatory networks with time-varying delays. IEEE Trans Circuits Syst I: Regul Pap 71(4–6):834–842

    Google Scholar 

  27. Ren F, Cao J (2008b) Exponential stability of discrete-time genetic regulatory networks with delays. IEEE Trans Neural Netw 19:520–523

    Google Scholar 

  28. Ren FL, Cao F, Cao JD (2015) Mittag–Leffler stability and generalized Mittag–Leffler stability of fractional-order gene regulatory networks. Neurocomputing 160(21):185–190

    Google Scholar 

  29. Song YF, Sum W, Jiang F (2016) Mean-square exponential input-to-state stability for neutral stochastic neural networks with mixed delays. Neurocomputing 205:195–203

    Google Scholar 

  30. Sun YH, Feng G, Cao JD (2010) A new approach to dynamic fuzzy modeling of genetic regulatory networks. Neural Process Lett 9(4):263–272

    Google Scholar 

  31. Tao B, Xiao M, Sun QS, Cao JD (2018) Hopf bifurcation analysis of a delayed fractional-order genetic regulatory network model. Neurocomputing 275:677–686

    Google Scholar 

  32. Vidyasagar M (2002) Nonlinear systems analysis. Prectice Hall, Upper Saddle River

    MATH  Google Scholar 

  33. Wang G, Cao J (2009) Robust exponential stability analysis for stochastic genetic networks with uncertain parameters. Commun Nonlinear Sci Numer Simul 14(8):3369–3378

    MathSciNet  MATH  Google Scholar 

  34. Wang Y, Ma ZJ, Shen JW, Liu ZR, Chen LN (2009) Periodic oscillation in delayed gene networks with SUM regulatory logic and small perturbations. Math Biosci 220(1):34–44

    MathSciNet  MATH  Google Scholar 

  35. Wang Y, Wang ZD, Liang JL, Li YR, Du M (2010) Synchronization of stochastic genetic oscillator networks with time delays and Markovian jumping parameters. Neurocomputing 73(13–15):2532–2539

    Google Scholar 

  36. Wang Z, Wu H, Liang J, Cao J, Liu X (2013) On modeling and state estimation for genetic regulatory networks with polytopic uncertainties. IEEE Trans NanoBiosci 12(1):13–20

    Google Scholar 

  37. Wu FX (2012) Stability and bifurcation of ring-structured genetic regulatory networks with time delays. IEEE Trans Circuits Syst I: Regul Pap 59(6):1312–1320

    MathSciNet  Google Scholar 

  38. Yousefi MR, Datta A, Doughert ER (2013) Optimal intervention in Markovian gene regulatory networks with random-length therapeutic response to antitumor drug. IEEE Trans Biomed Eng 60(12):3542–3552

    Google Scholar 

  39. Zhang D, Chen J, Park JH, Cao JD (2018a) Robust \(\cal{H}_{\infty }\) control for nonhomogeneous Markovian jump systems subject to quantized feedback and probabilistic measurements. J Frankl Inst 355(15):6992–7010

    MathSciNet  MATH  Google Scholar 

  40. Zhang HX, Hu J, Zou L, Yu XY, Wu ZH (2018b) Event-based state estimation for time-varying stochastic coupling networks with missing measurements under uncertain occurrence probabilities. Int J Gen Syst 47(5):506–521

    MathSciNet  Google Scholar 

  41. Zhang HX, Hu J, Liu HJ, Yu XY, Liu FQ (2019) Recursive state estimation for time-varying complex networks subject to missing measurements and stochastic inner coupling under random access protocol. Neurocomputing 346:506–521. https://doi.org/10.1016/j.neucom.2018.07.086

    Article  Google Scholar 

  42. Zhang Z, Li A, Yang L (2017) Input-to-state stability of memristor-based complex-valued neural networks with time delays. Neurocomputing 221:159–167

    Google Scholar 

  43. Zhang Z, Zhang J, Ai ZY (2018c) A novel stability criterion of the time-lag fractional-order gene regulatory network system for stability analysis. Commun Nonlinear Sci Numer Simul 66(2019):1019–1041

    MathSciNet  Google Scholar 

  44. Zhou L, Liu XT (2017) Mean-square exponential input-to-state stability of stochastic recurrent neural networks with multi-proportional delays. Neurocomputing 219:396–403

    Google Scholar 

  45. Zhu QX, Cao JD (2014) Mean-square exponential input-to-state stability of stochastic delayed neural networks. Neurocomputing 131:157–163

    Google Scholar 

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Acknowledgements

This work was jointly supported by the National Natural Science Foundation of China under Grant Nos. 61573291 and 61573096, and the Fundamental Research Funds for Central Universities XDJK2016B036.

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Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Guoxiong Xu & Haibo Bao

  2. School of Mathematics, Southeast University, Nanjing, 210096, People’s Republic of China

    Jinde Cao

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  1. Guoxiong Xu
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  2. Haibo Bao
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  3. Jinde Cao
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Correspondence to Haibo Bao.

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Xu, G., Bao, H. & Cao, J. Mean-Square Exponential Input-to-State Stability of Stochastic Gene Regulatory Networks with Multiple Time Delays. Neural Process Lett 51, 271–286 (2020). https://doi.org/10.1007/s11063-019-10087-9

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