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Traveling pulses and its wave solution scheme in a diffusively coupled 2D Hindmarsh-Rose excitable systems

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Abstract

In this article, an analytical approach is demonstrated to show the emerging traveling pulses for the local evolution of a set of diffusively coupled dynamical equations representing neuronal impulses. The derived dynamics governing the traveling pulses solution is described in a space-time reference frame with a two-dimensional excitable Hindmarsh-Rose (H-R) type oscillator. We deduce the conditions that allow us to describe explicitly the nature of propagating traveling pulses. We have constructed the detailed analytical results using semi-discrete approximation method with numerical simulations illuminating possible traveling pulses that include dispersion relations and group velocity equations. We show that the diffusive network can be expressed by the complex Ginzburg-Landau equation. The extended excitable medium with a homogeneous diffusive connection exhibits envelope solitons and multipulses. We observe how the series expansion parameter and coupling play key roles for the appearance of different traveling pulses. The transition phases and amplitude modulations are reported. The obtained results in the form of single solitary pulses and multipulses, reveal the possibility of collective behavior for information processing in excitable system.

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Data Availability Statement

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Acknowledgements

This work is supported by the National Board for Higher Mathematics (NBHM), Department of Atomic Energy, Govt. of India under Grant No. 02011/11/2022NBHM(R.P)/R & D-II/10217 to the author Argha Mondal.

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Authors and Affiliations

  1. Department of Mathematics, Sidho-Kanho-Birsha University, Purulia, West Bengal, 723104, India

    Subhashis Das, Madhurima Mukherjee, Argha Mondal & Sanat Kumar Mahato

  2. Department of Mathematical Sciences, University of Essex, Wivenhoe Park, Colchester, UK

    Argha Mondal

  3. Department of Mathematics, Ramakrishna Mission Vivekananda Centenary College, Rahara, West Bengal, 700118, India

    Kshitish Ch. Mistri

  4. Normandie Universite, UNIHAVRE, LMAH,FR-CNRS-3335, ISCN, 76600, Le Havre, France

    M. A. Aziz-Alaoui

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  1. Subhashis Das
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  2. Madhurima Mukherjee
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  6. M. A. Aziz-Alaoui
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Correspondence to Argha Mondal or M. A. Aziz-Alaoui.

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Appendix

Appendix

Equation 6 can be written in the following form

$$\begin{aligned}{} & {} \left( {\mathop \epsilon \nolimits ^2 \frac{{\mathop \partial \nolimits ^2 P}}{{\partial \mathop T\nolimits _1^2 }} - \epsilon 2i\omega \frac{{\partial P}}{{\partial \mathop T\nolimits _1 }} - \mathop \epsilon \nolimits ^2 2i\omega \frac{{\partial P}}{{\partial \mathop T\nolimits _2 }} - \mathop \omega \nolimits ^2 P} \right) \mathop e\nolimits ^{i\mathop \Omega \nolimits _k } \\{} & {} \quad + \left( {\mathop \epsilon \nolimits ^2 \frac{{\mathop \partial \nolimits ^2 \mathop P\nolimits ^* }}{{\partial \mathop T\nolimits _1^2 }} + \epsilon 2i\omega \frac{{\partial \mathop P\nolimits ^* }}{{\partial \mathop T\nolimits _1 }} + \mathop \epsilon \nolimits ^2 2i\omega \frac{{\partial \mathop P\nolimits ^* }}{{\partial \mathop T\nolimits _2 }} - \mathop \omega \nolimits ^2 \mathop P\nolimits ^* } \right) \\{} & {} \quad \mathop e\nolimits ^{ - i\mathop \Omega \nolimits _k } +\left( { - \mathop \epsilon \nolimits ^2 4i\omega \frac{{\partial R}}{{\partial \mathop T\nolimits _1 }} - \epsilon 4\mathop \omega \nolimits ^2 R} \right) \mathop e\nolimits ^{2i\mathop \Omega \nolimits _k } \\{} & {} \quad + \left( {\mathop \epsilon \nolimits ^2 4i\omega \frac{{\partial \mathop R\nolimits ^* }}{{\partial \mathop T\nolimits _1 }} - \epsilon 4\mathop \omega \nolimits ^2 \mathop R\nolimits ^* } \right) \mathop e\nolimits ^{ - 2i\mathop \Omega \nolimits _k } \\{} & {} \quad +\mathop \epsilon \nolimits ^2 \left[ {\left( {\frac{b}{c} - c} \right) + c\left( {\mathop P\nolimits ^2 \mathop e\nolimits ^{2i\mathop \Omega \nolimits _k } + \mathop P\nolimits ^{*2} \mathop e\nolimits ^{ - 2i\mathop \Omega \nolimits _k } + 2P\mathop P\nolimits ^* } \right) } \right] \\{} & {} \quad \left( { - i\omega P\mathop e\nolimits ^{i\mathop \Omega \nolimits _k } + i\omega \mathop P\nolimits ^* \mathop e\nolimits ^{ - i\mathop \Omega \nolimits _k } } \right) + \left( {d - b} \right) \\{} & {} \quad \left[ {P\mathop e\nolimits ^{i\mathop \Omega \nolimits _k } + \mathop P\nolimits ^* \mathop e\nolimits ^{ - i\mathop \Omega \nolimits _k } + \epsilon \left( {Q + R\mathop e\nolimits ^{2i\mathop \Omega \nolimits _k } + \mathop R\nolimits ^* \mathop e\nolimits ^{ - 2i\mathop \Omega \nolimits _k } } \right) } \right] \\{} & {} \quad +\epsilon \left( {\mathop P\nolimits ^2 \mathop e\nolimits ^{2i\mathop \Omega \nolimits _k } + \mathop P\nolimits ^{*2} \mathop e\nolimits ^{ - 2i\mathop \Omega \nolimits _k } + 2P\mathop P\nolimits ^* } \right) \\{} & {} \quad + 2\mathop \epsilon \nolimits ^2 \left( {P\mathop e\nolimits ^{i\mathop \Omega \nolimits _k } + \mathop P\nolimits ^* \mathop e\nolimits ^{ - i\mathop \Omega \nolimits _k } } \right) \left( {Q + R\mathop e\nolimits ^{2i\mathop \Omega \nolimits _k } + \mathop R\nolimits ^* \mathop e\nolimits ^{ - 2i\mathop \Omega \nolimits _k } } \right) \\{} & {} \quad +\frac{b}{3}\mathop \epsilon \nolimits ^2 \left( \mathop P\nolimits ^3 \mathop e\nolimits ^{3i\mathop \Omega \nolimits _k } + \mathop P\nolimits ^{*3} \mathop e\nolimits ^{ - 3i\mathop \Omega \nolimits _k }\right. \\{} & {} \quad \left. + 3\mathop P\nolimits ^2 \mathop P\nolimits ^* \mathop e\nolimits ^{i\mathop \Omega \nolimits _k } + 3P\mathop P\nolimits ^{*2} \mathop e\nolimits ^{ - 2i\mathop \Omega \nolimits _k } \right) + o\left( {\mathop \epsilon \nolimits ^3 } \right) \\{} & {} = \frac{b}{c}D\,( \left( {P + \epsilon \frac{{\partial P}}{{\partial \mathop X\nolimits _1 }} + \mathop \epsilon \nolimits ^2 \frac{{\partial P}}{{\partial \mathop X\nolimits _2 }} + \frac{{\mathop \epsilon \nolimits ^2 }}{2}\frac{{\mathop \partial \nolimits ^2 P}}{{\partial \mathop X\nolimits _1^2 }}} \right) \mathop e\nolimits ^{iq} \mathop e\nolimits ^{i\mathop \Omega \nolimits _k }\\{} & {} \quad +\left( {\mathop P\nolimits ^* + \epsilon \frac{{\partial \mathop P\nolimits ^* }}{{\partial \mathop X\nolimits _1 }} + \mathop \epsilon \nolimits ^2 \frac{{\partial \mathop P\nolimits ^* }}{{\partial \mathop X\nolimits _2 }} + \frac{{\mathop \epsilon \nolimits ^2 }}{2}\frac{{\mathop \partial \nolimits ^2 \mathop P\nolimits ^* }}{{\partial \mathop X\nolimits _1^2 }}} \right) \mathop e\nolimits ^{ - iq} \mathop e\nolimits ^{ - i\mathop \Omega \nolimits _k } \\{} & {} \quad +\epsilon \left( Q + \epsilon \frac{{\partial Q}}{{\partial \mathop X\nolimits _1 }} +\left( {R + \epsilon \frac{{\partial R}}{{\partial \mathop X\nolimits _1 }}} \right) \mathop e\nolimits ^{2iq} \mathop e\nolimits ^{2i\mathop \Omega \nolimits _k }\right. \\{} & {} \quad \left. + \left( {\mathop R\nolimits ^* + \epsilon \frac{{\partial \mathop R\nolimits ^* }}{{\partial \mathop X\nolimits _1 }}} \right) \mathop e\nolimits ^{ - 2iq} \mathop e\nolimits ^{ - 2i\mathop \Omega \nolimits _k } \right) \\{} & {} \quad -2\left( {P\mathop e\nolimits ^{i\mathop \Omega \nolimits _k } + \mathop P\nolimits ^* \mathop e\nolimits ^{ - i\mathop \Omega \nolimits _k } + \epsilon \left( {Q + R\mathop e\nolimits ^{2i\mathop \Omega \nolimits _k } + \mathop R\nolimits ^* \mathop e\nolimits ^{ - 2i\mathop \Omega \nolimits _k } } \right) } \right) \\{} & {} \quad +\left( {P - \epsilon \frac{{\partial P}}{{\partial \mathop X\nolimits _1 }} - \mathop \epsilon \nolimits ^2 \frac{{\partial P}}{{\partial \mathop X\nolimits _2 }} + \frac{{\mathop \epsilon \nolimits ^2 }}{2}\frac{{\mathop \partial \nolimits ^2 P}}{{\partial \mathop X\nolimits _1^2 }}} \right) \mathop e\nolimits ^{ - iq} \mathop e\nolimits ^{i\mathop \Omega \nolimits _k } \\{} & {} \quad +\left( {\mathop P\nolimits ^* - \epsilon \frac{{\partial \mathop P\nolimits ^* }}{{\partial \mathop X\nolimits _1 }} - \mathop \epsilon \nolimits ^2 \frac{{\partial \mathop P\nolimits ^* }}{{\partial \mathop X\nolimits _2 }} + \frac{{\mathop \epsilon \nolimits ^2 }}{2}\frac{{\mathop \partial \nolimits ^2 \mathop P\nolimits ^* }}{{\partial \mathop X\nolimits _1^2 }}} \right) \mathop e\nolimits ^{iq} \mathop e\nolimits ^{ - i\mathop \Omega \nolimits _k } \\{} & {} \quad +\epsilon \Bigg ( {Q - \epsilon \frac{{\partial Q}}{{\partial \mathop X\nolimits _1 }} + \Bigg ( {R - \epsilon \frac{{\partial R}}{{\partial \mathop X\nolimits _1 }}} \Bigg )\mathop e\nolimits ^{ - 2iq} \mathop e\nolimits ^{2i\mathop \Omega \nolimits _k }} \\{} & {} \quad + \Bigg ( {\mathop R\nolimits ^* - \epsilon \frac{{\partial \mathop R\nolimits ^* }}{{\partial \mathop X\nolimits _1 }}} \Bigg )\mathop e\nolimits ^{2iq} \mathop e\nolimits ^{ - 2i\mathop \Omega \nolimits _k} \Bigg ) \Bigg ) \\{} & {} \quad +i\omega \mathop \epsilon \nolimits ^2 D( - P\mathop e\nolimits ^{iq} \mathop e\nolimits ^{i\mathop \Omega \nolimits _k } + \mathop P\nolimits ^* \mathop e\nolimits ^{ - iq} \mathop e\nolimits ^{ - i\mathop \Omega \nolimits _k }\\{} & {} \quad - P\mathop e\nolimits ^{ - iq} \mathop e\nolimits ^{i\mathop \Omega \nolimits _k } + \mathop P\nolimits ^* \mathop e\nolimits ^{iq} \mathop e\nolimits ^{ - i\mathop \Omega \nolimits _k } )\\{} & {} \quad -2i\omega \mathop \epsilon \nolimits ^2 D\left( { - P\mathop e\nolimits ^{i\mathop \Omega \nolimits _k } + \mathop P\nolimits ^* \mathop e\nolimits ^{ - i\mathop \Omega \nolimits _k } } \right) + o\left( {\mathop \epsilon \nolimits ^3 } \right) , \end{aligned}$$

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Das, S., Mukherjee, M., Mondal, A. et al. Traveling pulses and its wave solution scheme in a diffusively coupled 2D Hindmarsh-Rose excitable systems. Nonlinear Dyn 111, 6745–6755 (2023). https://doi.org/10.1007/s11071-022-08168-x

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