Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane

doi:10.1088/1674-1056/ac9cbe

Abstract Noether theorem is applied to a variable order fractional multiscale mechano-electrophysiological model of neuron membrane dynamics. The variable orders fractional Lagrange equation of a multiscale mechano-electrophysiological model of neuron membrane dynamics is given. The variable orders fractional Noether symmetry criterion and Noether conserved quantities are given. The forms of variable orders fractional Noether conserved quantities corresponding to Noether symmetry generators solutions of the model under different conditions are discussed in detail, and it is found that the expressions of variable orders fractional Noether conserved quantities are closely dependent on the external nonconservative forces and material parameters of the neuron.

Keywords: Hamilton's principle Noether theorem fractional derivative multiscale electromechanical coupling neuron membrane

Received: 14 August 2022
Revised: 29 September 2022
Accepted manuscript online: 21 October 2022

PACS:	45.20.Jj	(Lagrangian and Hamiltonian mechanics)
	11.30.-j	(Symmetry and conservation laws)
	45.10.Hj	(Perturbation and fractional calculus methods)
	46.70.Hg	(Membranes, rods, and strings)

Fund: Project supported by the National Natural Science Foundation of China (Grant Nos. 12272148 and 11772141).

Corresponding Authors: Peng Wang
E-mail: cea_wangp@ujn.edu.cn

Cite this article:

Peng Wang(王鹏) Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane 2023 Chin. Phys. B 32 074501

[1] Hodgkin A and Huxley A A 1952 J. Physiol. 117 500
[2] Heimburg T and Jackson A D 2005 Proc. Natl. Acad. Sci. USA 102 9790
[3] Jérusalem A, Al-Rekabi Z, Chen H Y, et al. 2019 Acta Biomaterialia 97 116
[4] Engelbrecht J, Tamm T and Peets T 2021 Proceedings of the Estonian Academy of Sciences 70 3
[5] Chen H Y, Garcia-Gonzalez D and Jérusalem A 2019 Phys. Rev. E 99 032406
[6] Drapaca C S 2015 Frontiers in Cellular Neuroscience 9 271
[7] Drapaca C S 2017 J. Mech. Materials and Structures 12 35
[8] Bluman G W and Anco S C 2002 Symmetry and Integration Methods for Differential Equations (Berlin: Springer-Verlag)
[9] Noether E 1918 Nachr. Akad. Wiss. Gött Math-Phys. Kl 235
[10] Mei F X 2004 Symmetries and Conserved Quantities of Constrained Mechanical Systems (Beijing: Beijing Institute of Technology Press) (in Chinese)
[11] Wu H B and Mei F X 2005 Acta Phys. Sin. 54 2474 (in Chinese)
[12] Lou Z M 2005 Acta Phys. Sin. 54 1460 (in Chinese)
[13] Luo S K 2007 Chin. Phys. 16 3182
[14] Fu J L, Chen B Y and Chen L Q 2009 Phys. Lett. A 373 409
[15] Liu C, Zhao Y H and Chen X W 2010 Acta Phys. Sin. 59 11 (in Chinese)
[16] Zhang W W and Fang J H 2012 Chin. Phys. B 21 110201
[17] Wang P 2011 Chin. Phys. Lett. 28 040203
[18] Wang P 2012 Nonlinear Dyn. 68 53
[19] Wang P and Zhu H J 2011 Acta Phys. Pol. A 119 298
[20] Jiang W A, Liu K, Zhao G L and Chen M 2018 Acta Mech. 229 4771
[21] Dorodnitsyn V A and Ibragimov N H 2014 Commun. Nonlinear Sci. Numer. Simulat. 19 328
[22] Zhang Y 2018 Int. J. Non-Linear Mech. 101 36
[23] Riewe F 1997 Phys. Rev. E 55 3581
[24] Agrawal O P 2002 J. Math. Anal. Appl. 272 368
[25] Frederico F S F and Torres D F M 2007 J. Math. Anal. Appl. 334 834
[26] Atanackovic T M, Konjik S, Pilipovic S and Simic S 2009 Nonlinear Analysis 71 1504
[27] Zhou Y and Zhang Y 2014 Chin. Phys. B 23 124502
[28] Song C J and Cheng Y 2021 Adv. Math. Phys. 2021 1959643
[29] Zhang S H, Chen B Y and Fu J L 2012 Chin. Phys. B 21 100202
[30] Luo S K and Xu Y L 2015 Acta Mech. 226 829
[31] Luo S K and Li L 2013 Nonlinear Dyn. 73 339
[32] Zhang H B and Chen H B 2016 Nonlinear Dyn. 83 347
[33] Zhang H B and Chen H B 2018 Int. J. Non-Linear Mech. 107 34
[34] Atanackovic T M and Pilipovic S 2011 Fract. Calc. Appl. Anal. 14 94
[35] Chicone C and Mashhoon B 2002 Ann. Phys. 514 309
[36] Coimbra C F M 2003 Ann. Phys. 515 692

[1]	Conformable fractional heat equation with fractional translation symmetry in both time and space W S Chung, A Gungor, J Kříž, B C Lütfüoǧlu, and H Hassanabadi. Chin. Phys. B, 2023, 32(4): 040202.
[2]	A local refinement purely meshless scheme for time fractional nonlinear Schrödinger equation in irregular geometry region Tao Jiang(蒋涛), Rong-Rong Jiang(蒋戎戎), Jin-Jing Huang(黄金晶), Jiu Ding(丁玖), and Jin-Lian Ren(任金莲). Chin. Phys. B, 2021, 30(2): 020202.
[3]	First integrals of the axisymmetric shape equation of lipid membranes Yi-Heng Zhang(张一恒), Zachary McDargh, Zhan-Chun Tu(涂展春). Chin. Phys. B, 2018, 27(3): 038704.
[4]	Stochastic bifurcations of generalized Duffing-van der Pol system with fractional derivative under colored noise Wei Li(李伟), Mei-Ting Zhang(张美婷), Jun-Feng Zhao(赵俊锋). Chin. Phys. B, 2017, 26(9): 090501.
[5]	Stochastic response of van der Pol oscillator with two kinds of fractional derivatives under Gaussian white noise excitation Yong-Ge Yang(杨勇歌), Wei Xu(徐伟), Ya-Hui Sun(孙亚辉), Xu-Dong Gu(谷旭东). Chin. Phys. B, 2016, 25(2): 020201.
[6]	Non-Noether symmetries of Hamiltonian systems withconformable fractional derivatives Lin-Li Wang (王琳莉) and Jing-Li Fu(傅景礼). Chin. Phys. B, 2016, 25(1): 014501.
[7]	Response of a Duffing—Rayleigh system with a fractional derivative under Gaussian white noise excitation Zhang Ran-Ran (张冉冉), Xu Wei (徐伟), Yang Gui-Dong (杨贵东), Han Qun (韩群). Chin. Phys. B, 2015, 24(2): 020204.
[8]	Space–time fractional KdV–Burgers equation for dust acoustic shock waves in dusty plasma with non-thermal ions Emad K. El-Shewy, Abeer A. Mahmoud, Ashraf M. Tawfik, Essam M. Abulwafa, Ahmed Elgarayhi. Chin. Phys. B, 2014, 23(7): 070505.
[9]	Noether's theorems of a fractional Birkhoffian system within Riemann–Liouville derivatives Zhou Yan (周燕), Zhang Yi (张毅). Chin. Phys. B, 2014, 23(12): 124502.
[10]	Fractional cyclic integrals and Routh equations of fractional Lagrange system with combined Caputo derivatives Wang Lin-Li (王琳莉), Fu Jing-Li (傅景礼). Chin. Phys. B, 2014, 23(12): 124501.
[11]	Fractional Cattaneo heat equation in a semi-infinite medium Xu Huan-Ying (续焕英), Qi Hai-Tao (齐海涛), Jiang Xiao-Yun (蒋晓芸). Chin. Phys. B, 2013, 22(1): 014401.
[12]	Fractional differential equations of motion in terms of combined Riemann–Liouville derivatives Zhang Yi (张毅). Chin. Phys. B, 2012, 21(8): 084502.
[13]	Fractional charges and fractional spins for composite fermions in quantum electrodynamics Wang Yong-Long(王永龙), Lu Wei-Tao(卢伟涛), Jiang Hua(蒋华) Xu Chang-Tan(许长谭), and Pan Hong-Zhe(潘洪哲) . Chin. Phys. B, 2012, 21(7): 070501.
[14]	Hamilton formalism and Noether symmetry for mechanico–electrical systems with fractional derivatives Zhang Shi-Hua (张世华), Chen Ben-Yong (陈本永), Fu Jing-Li (傅景礼). Chin. Phys. B, 2012, 21(10): 100202.
[15]	The approximate conserved quantity of the weakly nonholonomic mechanical-electrical system Liu Xiao-Wei(刘晓巍), Li Yuan-Cheng(李元成), and Xia Li-Li(夏丽莉) . Chin. Phys. B, 2011, 20(7): 070203.

No Suggested Reading articles found!

Viewed

Full text

Abstract

Cited

Metrics
Related Articles

Fractional Noether theorem and fractional Lagrange equation of multi-scale mechano-electrophysiological coupling model of neuron membrane

Cite this article:

Online attention