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Combined action of time delay and colored cross-correlated Gaussian colored noises on dynamical characteristics for a FitzHugh–Nagumo neural system

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Abstract

In this paper, we focus on the investigation of the regime shift of the steady states, the mean first-passage time (MFPT) and the stochastic resonance (SR) for a FitzHugh–Nagumo neural system with time delay perturbed by colored cross-correlated Gaussian colored noises as well as a periodic signal. By means of a series of numerical calculations, our investigation results show that time delay, the multiplicative noise and the additive one can all produce the negative influence on the maintenance of the stability for the neuronal system; while the two self-correlation times of the internal and external noises, the noise correlation strength and its correlation time can always strengthen the stability of the biological system. As for the MFPT for the FHN system, it is observed that during the recovery process from the excited state to the resting one, we should take measures to increases the noise correlation strength, two noise self-correlation times, and reduce time delay along with the noise correlation time so as to sustain the state of excitation for neuronal cells as far as possible. With respect to the SR phenomenon, it is observed that the noise correlation strength and its correlation time, two Gaussian noise correlation times \(\tau_{1}\), \(\tau_{2}\) can all amplify significantly the SR effect, and even stimulate the double-peaked or three-peaked phenomenon; While time delay and the additive noise will always reduce the SR effect.

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Funding

Project is supported by the National Natural Science Foundation of China (Grant Nos. 61773012, 61371114), Six Talent Peaks Foundation Funded Project of Jiangsu Province (Grant No. JY-082), Jiangsu Provincial “Qing-Lan Engineering” Foundation Funded Project, the Jiangsu Provincial Key Laboratory of Networked Collective Intelligence under (Grant No. BM2017002), China Postdoctoral Science Foundation Funded Project (Grant No. 2016M591737), and Doctoral Research Startup Project of Jiangsu University of Science and Technology, China (Grant No. 1052931704).

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Authors and Affiliations

  1. School of Science, Jiangsu University of Science and Technology, Zhenjiang, 212003, China

    Kang-Kang Wang, Hui Ye, Ya-Jun Wang & Sheng-Hong Li

  2. Center of Complex Systems and Network Science Research, Southeast University, Nanjing, 210096, China

    Kang-Kang Wang

  3. School of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing, 210016, China

    Hui Ye

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  1. Kang-Kang Wang
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  2. Hui Ye
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Correspondence to Kang-Kang Wang.

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Appendix

Appendix

The coefficients of the modified potential \(U(x)\) are listed as follows:

\(d = 1 + (1 + a\beta )\tau \cdot \left\{ {(a - 2v_{2} )(1 - v_{2} ) - v_{2} (a - v_{2} ) + \frac{b}{\gamma }} \right\}\),

\(d_{1,2} = 1 + (1 + a\beta )\tau_{1,2} \cdot \left\{ {(a - 2v_{2} )(1 - v_{2} ) - v_{2} (a - v_{2} ) + \frac{b}{\gamma }} \right\}\),

$$ \begin{aligned} A_{1} &= - \frac{1}{2}\frac{{d^{2} \lambda^{2} d_{1}^{2} }}{1 + a\beta } + \frac{1}{2}\frac{{d_{2} dd_{1} a\lambda \sqrt {QM} \sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} }}{(1 + a\beta )QM} - \frac{1}{2}\frac{{d_{2}^{2} d\lambda^{2} d_{1}^{2} a}}{1 + a\beta } \hfill \\ &\quad + \frac{1}{2}\frac{{d_{2} d^{3} d_{1}^{{}} }}{1 + a\beta } + \frac{1}{2}\frac{{d_{2}^{{}} d^{3} d_{1}^{{}} a}}{1 + a\beta } - \frac{1}{2}\frac{{d_{2}^{2} d\lambda^{2} d_{1}^{2} }}{1 + a\beta } - \frac{1}{2}\frac{{d_{2} dd_{1} \lambda \sqrt {QM} \sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} }}{(1 + a\beta )QM} \hfill \\ &\quad + \frac{1}{2}\frac{{d_{2}^{2} d^{3} d_{1}^{{}} }}{1 + a\beta } - \frac{1}{2}\frac{{d_{2}^{2} d\lambda d_{1}^{2} a}}{1 + a\beta } - \frac{1}{2}\frac{{d_{2}^{2} dd_{1}^{2} \lambda \sqrt {QM} \sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} }}{(1 + a\beta )QM} \hfill \\ &\quad + \frac{1}{2}\frac{{d_{2}^{{}} d^{3} d_{1}^{2} a}}{1 + a\beta } + \frac{1}{2}\frac{{d_{2}^{{}} d^{2} d_{1}^{{}} \lambda \sqrt {QM} \sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} }}{(1 + a\beta )QM} \hfill \\ &\quad+ \frac{1}{2}\frac{{d_{2}^{2} d^{2} d_{1}^{2} a\lambda }}{1 + a\beta } + \frac{1}{2}\frac{{d_{2}^{2} d^{2} d_{1}^{{}} \lambda a\sqrt {QM} \sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} }}{(1 + a\beta )QM} - \frac{1}{2}\frac{{d_{2}^{{}} d^{2} \lambda^{3} d_{1}^{{}} }}{1 + a\beta } \hfill \\ &\quad+ \frac{1}{2}\frac{{d_{2}^{{}} d^{2} \sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} A\cos (\omega t)}}{(1 + a\beta )M} - \frac{1}{2}\frac{{d_{2}^{{}} d^{2} \sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} A\cos (\omega t)}}{(1 + a\beta )QM} \hfill \\ \end{aligned} $$
$$ \begin{aligned} A_{2} &= \frac{1}{2}d^{2} + \frac{1}{4}\frac{{db\sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} }}{(1 + a\beta )QM} + \frac{1}{4}\frac{{da\sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} }}{(1 + a\beta )QM} \hfill \\ &\quad - \frac{1}{4}\frac{{d_{1} \lambda \sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} \sqrt {QM} }}{{(1 + a\beta )Q^{2} M}} + \frac{1}{4}\frac{{d_{2} \lambda^{2} d_{1}^{2} }}{(1 + a\beta )Q} - \frac{1}{4}\frac{{d^{2} d_{1}^{{}} }}{(1 + a\beta )Q} - \frac{1}{2}\frac{{d_{2} \lambda^{2} d_{1}^{{}} }}{(1 + a\beta )QM} \hfill \\ \end{aligned} $$
$$ \begin{aligned} A_{3} &= \frac{1}{2}d^{2} + \frac{1}{4}\frac{{d_{2} \lambda^{2} d_{1}^{2} }}{(1 + a\beta )Q} - \frac{1}{4}\frac{{dd_{1}^{2} }}{(1 + a\beta )Q} - \frac{1}{4}\frac{{db\sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} }}{(1 + a\beta )QM} \hfill \\ &\quad + \frac{1}{4}\frac{{d_{1} \lambda \sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} \sqrt {QM} }}{{(1 + a\beta )Q^{2} M}} - \frac{1}{2}\frac{{d_{2} \lambda^{2} d_{1}^{{}} }}{(1 + a\beta )QM} - \frac{1}{4}\frac{{d\sqrt {QMd_{1} d_{2} (\lambda^{2} d_{1} d_{2} - d^{2} )} a}}{(1 + a\beta )QM} \hfill \\ \end{aligned} $$

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Wang, KK., Ye, H., Wang, YJ. et al. Combined action of time delay and colored cross-correlated Gaussian colored noises on dynamical characteristics for a FitzHugh–Nagumo neural system. Indian J Phys 96, 1943–1961 (2022). https://doi.org/10.1007/s12648-021-02186-y

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