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Dynamical Behaviors in Coupled FitzHugh-Nagumo Neural Systems with Time Delays

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Proceedings of 2016 Chinese Intelligent Systems Conference (CISC 2016)

Part of the book series: Lecture Notes in Electrical Engineering ((LNEE,volume 405))

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Abstract

It is observed that neuron encodes and integrates information employing a variety of complex dynamical behavior, such as spiking, bursting, periodicity, quasi-periodicity, and chaos. Time delay is an inevitable factor in the signal transmission between neurons, and neural system may lose its stability even for very small delay. In this paper, a model of coupled FitzHugh-Nagumo (FHN) neural system with two different delays is formulated, and its nonlinear dynamic behaviors such as stability, bifurcations, and chaos are then studied. It is shown that time delays can affect the stability of equilibrium states, and thereby lead to Hopf bifurcation and oscillation behavior. Moreover, some complex dynamics including quasi-periodic solutions and chaos are numerically demonstrated. Subsequently, numerical examples illustrate the effectiveness and feasibility of the theoretical results.

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References

  1. Hopfield JJ (1982) Neural networks and physical systems with emergent collective computational properties. Proc Nat Acad Sci 79:2554–2558

    Article  MathSciNet  Google Scholar 

  2. Hopfield JJ (1984) Neuronswith graded response have collective computational properties like those of two state neurons. Proc Nat Acad Sci 81:3088–3092

    Article  Google Scholar 

  3. Hopfield JJ, Tank DW (1986) Computing with neural circuits: a model. Science 233:625–633

    Article  Google Scholar 

  4. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane current and its application to conduction and excitation in nerve. J Physiol 117(4):500–544

    Article  Google Scholar 

  5. Hodgkin AL, Huxley AF (1952) A quantitative description of membrane and its application to conduction and excitation in nerve. J Physiol 117:500–44

    Article  Google Scholar 

  6. FitzHugh R (1961) Impulses and physiological state in theoretical models of nerve membrane. Biophys J 1:445–466

    Google Scholar 

  7. Nagumo J, Arimoto S, Yoshizawa S (1962) An active pulse transmission line simulating nerve axon. Proc IRE 50:2061–2070

    Google Scholar 

  8. Kim S, Park SH, Ryu CS (1997) Multistability in coupled oscillator systems with time delay. Phys Rev Lett 79:2911–2914

    Google Scholar 

  9. Bautin AN (1975) Qualitative investigation of a particular nonlinear system. J Appl Math Mech 39(4):606–615

    Google Scholar 

  10. Ueta T, Miyazaki H, Kousaka T, Kawakami H (2004) Bifurcation and chaos in coupled BVP oscillators. Int J Bifurcat Chaos Appl Sci Eng 14(4):1305–1324

    Article  MATH  Google Scholar 

  11. Ueta T, Kawakami H (2003) Bifurcation in asymmetrically coupled BVP oscillators. Int J Bifurcat Chaos Appl Sci Eng 13(5):1319–1327

    Article  MathSciNet  MATH  Google Scholar 

  12. Tsuji S, Ueta T, Kawakami H (2007) Bifurcation analysis of current coupled BVP oscillators. Int J Bifurcat Chaos Appl Sci Eng 17(3):837–850

    Article  MathSciNet  MATH  Google Scholar 

  13. Medetov B, Wei RC (2015) Numerically induced bursting in a set of coupled neuronal oscillators. Commun Nonlinear Sci Numer Simulat 20:1090–1098

    Google Scholar 

  14. Li YQ, Jiang WH (2011) Hopf and Bogdanov-Takens bifurcations in a coupled FitzHugh-Nagumo neural system with delay. Nonlinear Dyn 65:161–173

    Google Scholar 

  15. Jia JY, Liu HH, Xu CL, Yan F (2015) Dynamic effects of time delay on a coupled FitzHugh-Nagumo neural system. Alexandria Eng J 54:241–250

    Google Scholar 

  16. Xu X, Hu HY, Wang HL (2006) Stability switches, Hopf bifurcation and chaos of a neuron model with delay-dependent parameters. Phys Lett A 354:126–136

    Article  Google Scholar 

  17. Lu HT (2002) Chaotic attractors in delayed neural networks. Phys Lett A 298:109–116

    Article  MATH  Google Scholar 

  18. Yao SW, Tu HN (2014) Stability Switches and Hopf Bifurcation in a Coupled FitzHugh-Nagumo Neural System with Multiple Delays. Hindawi Publishing Corporation Abstract and Applied Analysis Vol 2014. Article ID 874701, 13 pp

    Google Scholar 

  19. Buric N, Todorovic D (2003) Dynamics of FitzHugh-Nagumo excitable systems with delayed coupling. Phys Rev E 67(Article ID 066222)

    Google Scholar 

  20. Fan D, Hong L (2010) Hopf bifurcation analysis in a synaptically coupled FHN neuron model with delays. Commun Nonlinear Sci Numer Simul 15(7):1873–1886

    Article  MathSciNet  MATH  Google Scholar 

  21. Ruan SG, Wei JJ (2003) On the zeros of transcendental functionals with applications to stability of delay differential equations with two delays. Dyn Contin Discrete Impuls Syst Ser A Math Anal 10:863–874

    Google Scholar 

Download references

Acknowledgments

The authors wish to thank the editor and the reviewers for their insightful and constructive comments, which improved this paper significantly. This work is supported by the National Science Foundation of China (No. 11272191).

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics, Shanghai University, 200072, Shanghai, China

    Yuan Zhang & Jin Zhou

  2. Department of Physics, School of Science, Shanghai University, 200444, Shanghai, China

    Lan Xiang

Authors
  1. Yuan Zhang
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  2. Lan Xiang
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  3. Jin Zhou
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Corresponding author

Correspondence to Jin Zhou .

Editor information

Editors and Affiliations

  1. Beihang University , Beijing, China

    Yingmin Jia

  2. Beijing University of Posts and Telecommunications, Beijing, China

    Junping Du

  3. University of Science and Technology Beijing, Beijing, China

    Weicun Zhang

  4. Tsinghua University , Beijing, China

    Hongbo Li

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© 2016 Springer Science+Business Media Singapore

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Zhang, Y., Xiang, L., Zhou, J. (2016). Dynamical Behaviors in Coupled FitzHugh-Nagumo Neural Systems with Time Delays. In: Jia, Y., Du, J., Zhang, W., Li, H. (eds) Proceedings of 2016 Chinese Intelligent Systems Conference. CISC 2016. Lecture Notes in Electrical Engineering, vol 405. Springer, Singapore. https://doi.org/10.1007/978-981-10-2335-4_28

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  • DOI: https://doi.org/10.1007/978-981-10-2335-4_28

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  • Publisher Name: Springer, Singapore

  • Print ISBN: 978-981-10-2334-7

  • Online ISBN: 978-981-10-2335-4

  • eBook Packages: Computer ScienceComputer Science (R0)

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