Geodesic
Modeling the Bursting Effect in Neuron Systems
Matematičeskie zametki, Tome 93 (2013) no. 5, pp. 684-701.

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We propose a new method for modeling the well-known phenomenon of “bursting behavior” in neuron systems by invoking delay equations. Namely, we consider a singularly perturbed nonlinear difference-differential equation with two delays describing the functioning of an isolated neuron. Under a suitable choice of parameters, we establish the existence of a stable periodic motion with any prescribed number of spikes on a closed time interval equal to the period length.
Keywords: “bursting behavior” in neuron systems, difference-differential equation, relay equation, Cauchy problem, Schauder principle, relaxation cycle, spiking, stability. 
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S. D. Glyzin; A. Yu. Kolesov; N. Kh. Rozov. Modeling the Bursting Effect in Neuron Systems. Matematičeskie zametki, Tome 93 (2013) no. 5, pp. 684-701. http://geodesic.mathdoc.fr/item/MZM_2013_93_5_a3/

[1] A. Yu. Kolesov, Yu. S. Kolesov, Relaksatsionnye kolebaniya v matematicheskikh modelyakh ekologii, Tr. MIAN, 199, Nauka, M., 1993 | MR | Zbl

[2] A. Yu. Kolesov, E. F. Mischenko, N. Kh. Rozov, “Rele s zapazdyvaniem i ego $C^1$-approksimatsiya”, Dinamicheskie sistemy i smezhnye voprosy, Sb. statei. K 60-letiyu so dnya rozhdeniya akademika Dmitriya Viktorovicha Anosova, Tr. MIAN, 216, Nauka, M., 1997, 126–153 | MR | Zbl

[3] A. Yu. Kolesov, N. Kh. Rozov, “Diskretnye avtovolny v sistemakh s zapazdyvaniem iz ekologii”, DAN, 434:6 (2010), 735–738 | MR | Zbl

[4] A. Yu. Kolesov, E. F. Mischenko, N. Kh. Rozov, “Ob odnoi modifikatsii uravneniya Khatchinsona”, Zh. vychisl. matem. i matem. fiz., 50:12 (2010), 2099–2112 | MR | Zbl

[5] A. Yu. Kolesov, N. Kh. Rozov, “Teoriya relaksatsionnykh kolebanii dlya uravneniya Khatchinsona”, Matem. sb., 202:6 (2011), 51–82 | DOI | MR | Zbl

[6] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Relaksatsionnye avtokolebaniya v neironnykh sistemakh. I”, Differents. uravneniya, 47:7 (2011), 919–932 | MR | Zbl

[7] E. F. Mischenko, N. Kh. Rozov, Differentsialnye uravneniya s malym parametrom i relaksatsionnye kolebaniya, Nauka, M., 1975 | MR | Zbl

[8] E. F. Mischenko, Yu. S. Kolesov, A. Yu. Kolesov, N. Kh. Rozov, Periodicheskie dvizheniya i bifurkatsionnye protsessy v singulyarno vozmuschennykh sistemakh, Fizmatlit, M., 1995 | MR | Zbl

[9] T. R. Chay, J. Rinzel, “Bursting, beating, and chaos in an excitable membrane model”, Biophys. J., 47:3 (1985), 357–366 | DOI

[10] G. B. Ermentrout, N. Kopell, “Parabolic bursting in an excitable system coupled with a slow oscillation”, SIAM J. Appl. Math., 46:2 (1986), 233–253 | DOI | MR | Zbl

[11] E. M. Izhikevich, “Neural excitability, spiking and bursting”, Int. J. Bifurcation Chaos, 10:6 (2000), 1171–1266 | DOI | MR | Zbl

[12] M. I. Rabinovich, P. Varona, A. I. Selverston, H. D. I. Abarbanel, “Dynamical principles in neuroscience”, Rev. Mod. Phys., 78:4 (2006), 1213–1265 | DOI

[13] Bursting. The Genesis of Rhythm in the Nervous System, eds. S. Coombes, P. C. Bressloff, World Sci. Publ., Hackensack, NJ, 2005 | MR | Zbl

[14] S. A. Kaschenko, V. V. Maiorov, Modeli volnovoi pamyati, Knizhnyi dom “LIBROKOM”, M., 2009

[15] A. Yu. Kolesov, E. F. Mischenko, N. Kh. Rozov, “Novye metody dokazatelstva suschestvovaniya i ustoichivosti periodicheskikh reshenii v singulyarno vozmuschennykh sistemakh s zapazdyvaniem”, Analiz i osobennosti. Chast 2, Sb. statei. K 70-letiyu so dnya rozhdeniya akademika Vladimira Igorevicha Arnolda, Tr. MIAN, 259, Nauka, M., 2007, 106–133 | MR | Zbl

[16] S. D. Glyzin, A. Yu. Kolesov, N. Kh. Rozov, “Ekstremalnaya dinamika obobschennogo uravneniya Khatchinsona”, Zh. vychisl. matem. i matem. fiz., 49:1 (2009), 76–89 | MR | Zbl