Geodesic
On Integrability of the FitzHugh – Rinzel Model
Russian journal of nonlinear dynamics, Tome 15 (2019) no. 1, pp. 13-19.

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The integrability of the FitzHugh – Rinzel model is considered. This model is an example of the system of equations having the expansion of the general solution in the Puiseux series with three arbitrary constants. It is shown that the FitzHugh – Rinzel model is not integrable in the general case, but there are two formal first integrals of the system of equations for its description. Exact solutions of the FitzHugh – Rinzel system of equations are given.
Keywords: FitzHugh – Rinzel model, first integral, general solution
Mots-clés : Painlevé test, exact solution. 
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N. A. Kudryashov. On Integrability of the FitzHugh – Rinzel Model. Russian journal of nonlinear dynamics, Tome 15 (2019) no. 1, pp. 13-19. http://geodesic.mathdoc.fr/item/ND_2019_15_1_a1/

[1] FitzHugh, R., “Impulses and Physiological States in Theoretical Models of Nerve Membrane”, Biophys. J., 1:6 (1961), 445–466 | DOI

[2] Nagumo, J., Arimoto, S., and Yoshizawa, S., “An Active Pulse Transmission Line Simulating Nerve Axon”, Proc. of the IRE, 50:10 (1962), 2061–2070 | DOI

[3] FitzHugh, R., “A Kinetic Model for the Conductance Changes in Nerve Membranes”, J. Cell. Comp. Physiol., 66:suppl. 2 (1965), 111–117 | DOI

[4] Hodgkin, A. L. and Huxley, A. F., “A Quantitative Description of Membrane Current and Its Application to Conduction and Excitation in Nerve”, J. Physiol., 117:4 (1952), 500–544 | DOI

[5] Postnikov, E. B. and Titkova, O. V., “A Correspondence between the Models of Hodgkin – Huxley and FitzHugh – Nagumo Revised”, Eur. Phys. J. Plus, 131:11 (2016), 411 | DOI

[6] Saha, A. and Feudel, U., “Extreme Events in FitzHugh – Nagumo Oscillators Coupled with Two Time Delays”, Phys. Rev. E, 95:6 (2017), 062219, 10 pp. | DOI | MR

[7] Schmidt, A., Kasmatis, Th., Hizanidas, J., Provata, A., and Hövel, P., “Chimera Patterns in Two-Dimensional Networks of Coupled Neurons”, Phys. Rev. E, 95:3 (2017), 032224, 13 pp. | DOI | MR

[8] Zemskov, E. P., Tsyganov, M. A., and Horsthemke, W., “Oscillatory Pulses and Wave Trains in a Bistable Reaction-Diffusion System with Cross Diffusion”, Phys. Rev. E, 95:1 (2017), 012203, 9 pp. | DOI | MR

[9] Scott, A., Nonlinear Science. Emergence and Dynamics of Coherent Structures, 2nd ed., Oxford Univ. Press, Oxford, 2005 | MR

[10] Kudryashov, N. A., “Asymptotic and Exact Solutions of the FitzHugh – Nagumo Model”, Regul. Chaotic Dyn., 23:2 (2018), 152–160 | DOI | MR | Zbl

[11] Kudryashov, N. A., Rybka, R. B., and Sboev, A. G., “Analytical Properties of the Perturbed FitzHugh – Nagumo Model”, Appl. Math. Lett., 76 (2018), 142–147 | DOI | MR | Zbl

[12] Llibre, J. and Vidal, C., “Periodic Solutions of a Periodic FitzHugh – Nagumo System”, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 25:13 (2015), 1550180, 6 pp. | DOI | MR | Zbl

[13] Rinzel, J., “A Formal Classification of Bursting Mechanisms in Excitable Systems”, Mathematical Topics in Population Biology, Morphogenesis and Neurosciences, Lecture Notes in Biomath., 71, eds. E. Teramoto, M. Yamaguti, Springer, Berlin, 1987, 267–281 | DOI | MR

[14] Izv. Vyssh. Uchebn. Zaved. Radiofizika, 49:11 (2006), 1002–1014 (Russian) | DOI

[15] Zemlyanukhin, A. I. and Bochkarev, A. V., “Analytical Properties and Solutions of the FitzHugh – Rinzel Model”, Russian J. Nonlinear Dyn., 15:1 (2019), 3–12 | MR

[16] Painlevé, P., “Sur les équations différentielles du second ordre et d'ordre supérieur dont l'intégrale générale est uniforme”, Acta Math., 25 (1902), 1–85 | DOI | MR

[17] Gambier, B., “Sur les équations différetielles dont l'integrate générale est uniforme”, C. R. Acad. Sci. Paris, 142 (1906), 266–269, 1403–1406, 1497–1500 | Zbl

[18] Kudryashov, N. A., “Amalgamations of the Painlevé Equations”, J. Math. Phys., 44:12 (2003), 6160–6178 | DOI | MR | Zbl

[19] Borisov, A. V. and Kudryashov, N. A., “Paul Painlevé and His Contribution to Science”, Regul. Chaotic Dyn., 19:1 (2014), 1–19 | DOI | MR | Zbl

[20] Kudryashov, N. A., “Higher Painlevé Transcensents As Special Solutions of Some Nonlinear Integrable Hierarchies”, Regul. Chaotic Dyn., 19:1 (2014), 48–63 | DOI | MR | Zbl

[21] Kudryashov, N. A., “Nonlinear Differential Equations Associated with the First Painlevé Hierarchy”, Appl. Math. Lett., 90 (2019), 223–228 | DOI | MR | Zbl

[22] Kudryashov, N. A., “Exact Solutions of the Equation for Surface Waves in a Convecting Fluid”, Appl. Math. Comput., 344/345 (2019), 97–106 | MR

[23] Kudryashov, N. A., “Exact Solutions and Integrability of the Duffing – van der Pol Equation”, Regul. Chaotic Dyn., 23:4 (2018), 471–479 | DOI | MR | Zbl

[24] Polyanin, A. D. and Zaitsev, V. F., Handbook of Exact Solutions for Ordinary Differential Equations, 2nd ed., Chapman Hall/CRC, Boca Raton, Fla., 2002 | MR

[25] Polyanin, A. D. and Zaitsev, V. F., Handbook of Nonlinear Partial Differential Equations, 2nd ed., CRC, Boca Raton, Fla., 2012 | MR

[26] Polyanin, A. D. and Zaitsev, V. F., Handbook of Ordinary Differential Equations: Exact Solutions, Methods, and Problems, Chapman Hall/CRC, Boca Raton, Fla., 2017, 1496 pp. | MR

[27] Kudryashov, N. A., “Exact Solutions of the Generalized Kuramoto – Sivashinsky Equation”, Phys. Lett. A, 147:5–6 (1990), 287–291 | DOI | MR

[28] Kudryashov, N. A., “Solitary and Periodic Solutions of the Generalized Kuramoto – Sivashinsky Equation”, Regul. Chaotic Dyn., 13:3 (2008), 234–238 | DOI | MR | Zbl

[29] Kudryashov, N. A., “One Method for Finding Exact Solutions of Nonlinear Differential Equations”, Commun. Nonlinear Sci. Numer. Simul., 17:6 (2012), 2248–2253 | DOI | MR | Zbl

[30] Kudryashov, N. A., “Polynomials in Logistic Function and Solitary Waves of Nonlinear Differential Equations”, Appl. Math. Comput., 219:17 (2013), 9245–9253 | MR | Zbl

[31] Kudryashov, N. A., “Painlevé Analysis and Exact Solutions of the Korteweg – de Vries Equation with a source”, Appl. Math. Lett., 41 (2015), 41–45 | DOI | MR | Zbl