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Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization

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Abstract

The primary objective of this paper is to propose a new approach for analyzing pinning stability in a complex dynamical network via impulsive control. A simple yet generic criterion of impulsive pinning synchronization for such coupled oscillator network is derived analytically. It is shown that a single impulsive controller can always pin a given complex dynamical network to a homogeneous solution. Subsequently, the theoretic result is applied to a small-world (SW) neuronal network comprised of the Hindmarsh–Rose oscillators. It turns out that the firing activities of a single neuron can induce synchronization of the underlying neuronal networks. This conclusion is obviously in consistence with empirical evidence from the biological experiments, which plays a significant role in neural signal encoding and transduction of information processing for neuronal activity. Finally, simulations are provided to demonstrate the practical nature of the theoretical results.

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Acknowledgements

This work was supported by the National Science Foundation of China (Grant Nos. 10972129 and 10832006), the Specialized Research Foundation for the Doctoral Program of Higher Education (Grant No. 200802800015), the Innovation Program of Shanghai Municipal Education Commission (Grant No. 10ZZ61), the Shanghai Leading Academic Discipline Project (Project No. S30106).

The authors sincerely thank the anonymous reviewers for their valuable comments that have led to the present improved version of the original manuscript.

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Authors and Affiliations

  1. Shanghai Institute of Applied Mathematics and Mechanics and Shanghai Key Laboratory of Mechanics in Energy Engineering, Shanghai University, Shanghai, 200072, China

    Jin Zhou

  2. School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 200090, China

    Quanjun Wu

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

    Lan Xiang

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  1. Jin Zhou
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  2. Quanjun Wu
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  3. Lan Xiang
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Correspondence to Jin Zhou.

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Zhou, J., Wu, Q. & Xiang, L. Impulsive pinning complex dynamical networks and applications to firing neuronal synchronization. Nonlinear Dyn 69, 1393–1403 (2012). https://doi.org/10.1007/s11071-012-0355-9

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  • DOI: https://doi.org/10.1007/s11071-012-0355-9

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