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Synchronization enhancement via an oscillatory bath in a network of self-excited cells

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Abstract

The possibility of using a dynamic environment to achieve and optimize phase synchronization in a network of self-excited cells with free-end boundary conditions is addressed in this paper. The dynamic environment is an oscillatory bath coupled linearly to a network of four cells. The boundaries of the stable solutions of the dynamical states as well as the ranges of coupling parameters leading to stability and instability of synchronization are determined. Numerical simulations are used to check the accuracy and to complement the result obtained from analytical treatment. The robustness of synchronization strategy is tested using a local and global injection of Gaussian white noise in the network. The control gain parameter of the bath coupling can modulate the occurrence of synchronization in the network without prior requirement of direct coupling among all the cells. The process of synchronization obtained through local injection is independent of the node at which noise is injected into the system. As compared to local injection, the global injection scheme increases the range of noise amplitude for which synchronization occurs in the network.

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Acknowledgements

Part of this work was completed during a research visit of Dr Nana Nbendjo at the Institute of Theoretical Physics-State University of São Paulo in Brazil. He is grateful to Brazilian Government (CNPq) for financial support within the project CNPq /PROAFRICA 490265 /2010-3.

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Authors and Affiliations

  1. Laboratory of Modelling and Simulation in Engineering and Biomimetics and Prototypes, University of Yaoundé I, P.O. Box 812, Yaoundé, Cameroon

    B R NANA NBENDJO & H G ENJIEU KADJI

  2. Monell Chemical Senses Center, 3500 Market Street, Philadelphia, PA, 19104, USA

    H G ENJIEU KADJI

  3. Instituto de Física Teórica, UNESP-Universidade Estadual Paulista, Rua Dr. Bento Teobaldo Ferraz 271, Bloco II, 01140-070, São Paulo, SP, Brazil

    HILDA A CERDEIRA

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  1. B R NANA NBENDJO
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  2. H G ENJIEU KADJI
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  3. HILDA A CERDEIRA
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Correspondence to HILDA A CERDEIRA.

Appendix

Appendix

The Kirchhoff’s laws for each of the oscillator are as follows:

Oscillator 1:

$$\begin{array}{@{}rcl@{}} && V_{1}-V_{2}=L_{C}\frac{\mathrm{d}I_{1}}{\mathrm{d}\tau}\\ && V_{5}-V_{1}=L_{15}\frac{\mathrm{d}I_{5}}{\mathrm{d}\tau}\\ && I_{5}=I_{5}^{\text{Osc}}+I_{1}. \end{array} $$

Oscillator 2:

$$\begin{array}{@{}rcl@{}} &&V_{2}-V_{3}=L_{C}\frac{\mathrm{d}I_{2}}{\mathrm{d}\tau}\\ &&V_{2}-V_{5}=L_{25}\frac{\mathrm{d}{I_{1}^{L}}}{\mathrm{d}\tau}\\ &&I_{1}={I_{1}^{L}}+I_{1}^{\text{Osc}}+I_{2}. \end{array} $$

Oscillator 3:

$$\begin{array}{@{}rcl@{}} &&V_{3}-V_{4}=L_{C}\frac{\mathrm{d}I_{3}}{\mathrm{d}\tau}\\ &&V_{3}-V_{5}=L_{35}\frac{\mathrm{d}{I_{2}^{L}}}{\mathrm{d}\tau}\\ && I_{2}=I_{3}+{I_{2}^{L}}+I_{2}^{\text{Osc}}. \end{array} $$

Oscillator 4:

$$\begin{array}{@{}rcl@{}} && V_{3}-V_{4}=L_{C}\frac{\mathrm{d}I_{3}}{\mathrm{d}\tau}\\ && V_{4}-V_{5}=L_{45}\frac{\mathrm{d}I_{4}}{\mathrm{d}\tau}\\ && I_{3}=I_{4}+I_{3}^{\text{Osc}}. \end{array} $$

Oscillator 5:

$$\begin{array}{@{}rcl@{}} && V_{4}-V_{5}=L_{45}\frac{\mathrm{d}I_{4}}{\mathrm{d}\tau}\\ &&V_{5}-V_{1}=L_{15}\frac{\mathrm{d}I_{5}}{\mathrm{d}\tau}\\ &&V_{2}-V_{5}=L_{25}\frac{\mathrm{d}{I_{1}^{L}}}{\mathrm{d}\tau}\\ &&V_{3}-V_{5}=L_{35}\frac{\mathrm{d}{I_{2}^{L}}}{\mathrm{d}\tau}\\ &&I_{4}=I_{5}+I_{4}^{\text{Osc}}-{I_{1}^{L}}-{I_{2}^{L}}. \end{array} $$

From the above equations, the dynamics of the network in the case of vdPol oscillators is described by the following equations:

$$\begin{array}{@{}rcl@{}} \frac{\mathrm{d}^{2}V_{1}}{\mathrm{d}\tau}&-&\frac{a_{1}}{C}\left(1\,-\,3\frac{a_{3}}{a_{1}}{V_{1}^{2}}\right)\frac{ \mathrm{d}V_{1}}{\mathrm{d}\tau}\,+\,\frac{1}{LC}V_{1} \,=\,\frac{1}{L_{C} C}(V_{2}\,-\,V_{1})\,+\,\frac{1}{L_{15}C}(V_{5}-V_{1}),\\ \frac{\mathrm{d}^{2}V_{2}}{\mathrm{d}\tau}&-&\frac{a_{1}}{C}\left(1-3\frac{a_{3}}{a_{1}}{V_{2}^{2}}\right)\frac{ \mathrm{d}V_{2}}{\mathrm{d}\tau}+\frac{1}{LC}V_{2} \\ &&=\frac{1}{L_{C} C}(V_{1}-2V_{2}+V_{3})+\frac{1}{L_{25}C}(V_{5}-V_{2}),\\ \frac{\mathrm{d}^{2}V_{3}}{\mathrm{d}\tau}&-&\frac{a_{1}}{C}\left(1-3\frac{a_{3}}{a_{1}}{V_{3}^{2}}\right)\frac{ \mathrm{d}V_{3}}{\mathrm{d}\tau}+\frac{1}{LC}V_{3} \\ &&=\frac{1}{L_{C} C}(V_{2}-2V_{3}+V_{4})+\frac{1}{L_{35}C}(V_{5}-V_{3}),\\ \frac{\mathrm{d}^{2}V_{4}}{\mathrm{d}\tau}&-&\frac{a_{1}}{C}\left(1-3\frac{a_{3}}{a_{1}}{V_{4}^{2}}\right)\frac{ \mathrm{d}V_{4}}{\mathrm{d}\tau}\,+\,\frac{1}{LC}V_{4} \,=\,\frac{1}{L_{C} C}(V_{3}\,-\,V_{4})\,+\,\frac{1}{L_{45}C}(V_{5}\,-\,V_{4}),\\ \frac{\mathrm{d}^{2}V_{5}}{\mathrm{d}\tau}&-&\frac{a_{1}}{C}\left(1-3\frac{a_{3}}{a_{1}}{V_{5}^{2}}\right)\frac{ \mathrm{d}V_{5}}{\mathrm{d}\tau}+\frac{1}{LC}V_{5} \\ &&\!\!\!\!\!\!\!=\frac{1}{L_{15}C}(V_{1}\,-\,V_{5})\,+\,\frac{1}{L_{25}C}(V_{2}\,-\,V_{5}{\kern-.5pt})\,+\,\frac{1}{L_{35}C}(V_{3}\!{\kern-.2pt}-{\kern-.2pt}\!V_{5})\,+\,\frac{1}{L_{45}C}(V_{4}\!{\kern-.2pt}-{\kern-.2pt}\!V_{5}), \end{array} $$

where

$$\begin{array}{@{}rcl@{}} && \mu=a_{1}\sqrt{\frac{L}{C}},\quad \omega^{2}=\frac{1}{LC},\quad V_{\kappa}=\sqrt{\frac{a_{1}}{3a_{3}}}x_{\kappa},\quad t=\omega\tau,\\ && K=\frac{L}{L_{C}},\quad G_{i}=\frac{L}{L_{i5}},\quad 1\leq i\leq 5, \quad L_{15}=L_{25}=L_{35}=L_{45}. \end{array} $$

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NANA NBENDJO, B.R., ENJIEU KADJI, H.G. & CERDEIRA, H.A. Synchronization enhancement via an oscillatory bath in a network of self-excited cells. Pramana - J Phys 84, 257–272 (2015). https://doi.org/10.1007/s12043-014-0895-2

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