### AIMS Mathematics

2023, Issue 2: 3322-3337. doi: 10.3934/math.2023171
Research article

# The existence of solitary wave solutions for the neuron model with conductance-resistance symmetry

• Received: 26 August 2022 Revised: 01 November 2022 Accepted: 07 November 2022 Published: 17 November 2022
• MSC : 35C07, 35C08, 37C29

• The neuron model with conductance-resistance symmetry was recently derived by Deng, which is similar to the Hodgkin-Huxley equation, referred to as CRS neuron model. In this paper, we will consider a 2-dimensional reduction model qualitatively similar to the FitzHugh-Nagumo equation. We first give the derivation of the CRS neuron model in propagating action potential. And then we prove the existence of solitary wave solution for the 2-dimensional reduced CRS neuron model by using phase diagram analysis.

Citation: Feifei Cheng, Ji Li, Qing Yu. The existence of solitary wave solutions for the neuron model with conductance-resistance symmetry[J]. AIMS Mathematics, 2023, 8(2): 3322-3337. doi: 10.3934/math.2023171

### Related Papers:

• The neuron model with conductance-resistance symmetry was recently derived by Deng, which is similar to the Hodgkin-Huxley equation, referred to as CRS neuron model. In this paper, we will consider a 2-dimensional reduction model qualitatively similar to the FitzHugh-Nagumo equation. We first give the derivation of the CRS neuron model in propagating action potential. And then we prove the existence of solitary wave solution for the 2-dimensional reduced CRS neuron model by using phase diagram analysis.

 [1] R. Casten, H. Cohen, P. Lagerstrom, Perturbation analysis of an approximation to Hodgkin-Huxley theory, Quart. Appl. Math., 32 (1975), 365–402. https://doi.org/10.1090/qam/445095 doi: 10.1090/qam/445095 [2] C. Conley, On traveling wave solutions of nonlinear diffusion equations, In: Dynamical systems, theory and applications, Berlin: Springer-Verlag, 2005,498–510. https://doi.org/10.1007/3-540-07171-7_13 [3] G. Carpenter, A geometric approach to singular perturbation problems with applications to nerve impulse equations, J. Differ. Equations, 23 (1977), 335–367. http://dx.doi.org/10.1016/0022-0396(77)90116-4 doi: 10.1016/0022-0396(77)90116-4 [4] P. Carter, B. Sandstede, Fast pulses with oscillatory tails in the FitzHugh-Nagumo system, SIAM J. Math. Anal., 47 (2015), 3393–3441. https://doi.org/10.1137/140999177 doi: 10.1137/140999177 [5] P. Carter, A. Scheel, Wave train selection by invasion fronts in the FitzHugh-Nagumo equation, Nonlinearity, 31 (2018), 5536–5572. https://doi.org/10.1088/1361-6544/aae1db doi: 10.1088/1361-6544/aae1db [6] B. Deng, The existence of infinitely many traveling front and back waves in the FitzHugh-Nagumo equations, SIAM J. Math. Anal., 22 (1991), 1631–1650. https://doi.org/10.1137/0522102 doi: 10.1137/0522102 [7] B. Deng, Neuron model with conductance-resistance sysmmetry, Phys. Lett. A, 383 (2019), 125976. https://doi.org/10.1016/j.physleta.2019.125976 doi: 10.1016/j.physleta.2019.125976 [8] R. FitzHugh, Impulses and physiological states in theoretical models of nerve membrane, Biophys. J., 1 (1961), 445–466. https://doi.org/10.1016/S0006-3495(61)86902-6 doi: 10.1016/S0006-3495(61)86902-6 [9] R. Field, W. Troy, The existence of solitary travelling wave solutions of a model of the Belousov-Zhabotinskii reaction, SIMA J. Appl. Math., 37 (1979), 561–587. https://doi.org/10.1137/0137042 doi: 10.1137/0137042 [10] J. Feroe, Traveling waves of infinitely many pulses in nerve equations, Math. Biosci., 55 (1981), 189–203. https://doi.org/10.1016/0025-5564(81)90095-X doi: 10.1016/0025-5564(81)90095-X [11] G. Faye, Existence and stability of traveling pulses in a neural field equation with synaptic depression, SIAM J. Appl. Dyn. Syst., 12 (2013), 2032–2067. https://doi.org/10.1137/130913092 doi: 10.1137/130913092 [12] G. Faye, A. Scheel, Existence of pulses in excitable media with nonlocal coupling, Adv. Math., 270 (2015), 400–456. https://doi.org/10.1016/j.aim.2014.11.005 doi: 10.1016/j.aim.2014.11.005 [13] J. Greenberg, A note on the Nagumo equation, Quart. J. Math., 24 (1973), 307–314. https://doi.org/10.1093/qmath/24.1.307 doi: 10.1093/qmath/24.1.307 [14] W. Gao, J. Wang, Existence of wavefronts and impulses to FitzHugh-Nagumo equations, Nonlinear Anal.-Theor., 57 (2004), 667–676. https://doi.org/10.1016/j.na.2004.03.009 doi: 10.1016/j.na.2004.03.009 [15] A. Gawlik, V. Vladimirov, S. Skurativskyi, Existence of the solitary wave solutions supported by the modified FitzHugh-Nagumo system, Nonlinear Anal. Model., 25 (2020), 482–501. https://doi.org/10.15388/namc.2020.25.16842 doi: 10.15388/namc.2020.25.16842 [16] S. Hastings, On the existence of homoclinic and periodic orbits for the FitzHugh-Nagumo equations, Quart. J. Math., 27 (1976), 123–134. https://doi.org/10.1093/qmath/27.1.123 doi: 10.1093/qmath/27.1.123 [17] S. Hastings, Single and multiple pulse waves for the FitzHugh-Nagumo equations, SIAM J. Appl. Math., 42 (1982), 247–260. https://doi.org/10.1137/0142018 doi: 10.1137/0142018 [18] A. Hodgkin, A. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol., 117 (1952), 500–544. https://doi.org/10.1113/jphysiol.1952.sp004764 doi: 10.1113/jphysiol.1952.sp004764 [19] C. Jones, N. Kopell, R. Langer, Construction of the FitzHugh-Nagumo pulse using differential forms, In: Patterns and dynamics in reactive media, New York: Springer, 1991,101–115. https://doi.org/10.1007/978-1-4612-3206-3_7 [20] R. Langer, Existence and uniqueness of pulse solutions to the FitzHugh-Nagumo equations, Ph. D Thesis, Northeastern University, 1980. [21] J. Moore, Excitation of the squid axon membrane in isosmotic potassium chloride, Nature, 183 (1959), 265–266. https://doi.org/10.1038/183265b0 doi: 10.1038/183265b0 [22] H. Khalil, Nonlinear systems, 3 Eds., Prentice Hall Press, 2002. [23] H. Mckean, Nagumo's equation, Adv. Math., 4 (1970), 209–223. https://doi.org/10.1016/0001-8708(70)90023-X [24] J. Nagumo, S. Arimoto, S. Yoshizawa, An active pulse transmission line simulating nerve axon, Proceedings of the IRE, 50 (1962), 2061–2070. https://doi.org/10.1109/JRPROC.1962.288235 doi: 10.1109/JRPROC.1962.288235 [25] J. Rauch, J. Smoller, Qualitative theory of FitzHugh-Nagumo equations, Adv. Math., 27 (1978), 12–44. https://doi.org/10.1016/0001-8708(78)90075-0 doi: 10.1016/0001-8708(78)90075-0 [26] A. Winfree, Stately rotating patterns of reaction and diffusion, In: Theoretical chemistry, New York: Academic Press, 1978, 1–51. https://doi.org/10.1016/B978-0-12-681904-5.50007-3
• ##### Reader Comments
• © 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
###### 通讯作者: 陈斌, bchen63@163.com
• 1.

沈阳化工大学材料科学与工程学院 沈阳 110142

1.8 3.4

## Metrics

Article views(1019) PDF downloads(93) Cited by(0)

Article outline

Figures(4)

## Other Articles By Authors

• On This Site
• On Google Scholar

/

DownLoad:  Full-Size Img  PowerPoint