Abstract
This paper reports bursting behavior and related bifurcations in a fractional order FitzHugh-Nagumo neuron model, by adding sub fast-slow system. We classify different bursters of the system consisting fold/Hopf via a fold/fold hysteresis loop, homoclinic/homolininc cycle-cycle, fold/homoclinic, homoclinic/Hopf via homoclinic/fold hysteresis loop. We determine stability and dynamical behaviors of the equilibria of the system by numerical simulations.
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References
K. Diethelm, The Analysis of Fractional Differential Equations (Springer, Berlin, Heidelberg, 2010).
A. I. Ehibilik, R. M. Borisyuk, and D. Roose, “Numerical bifurcation analysis of a model of coupled neural oscillators,” Int. Ser. Numer. Math. 104, 215–228 (1992).
G. B. Ermentrout and N. Kopell, “Parabolic bursting in an excitable system coupled with a slow oscillation,” SIAM J. Appl. Math. 46, 233–253 (1986).
G. B. Ermentrout, “Type I membranes, phase resetting curves and synchrony,” Neural Comput. 8, 979–1001 (1996).
R. Fitzhugh, “Impulses and physiological states in models of nerve membrane,” Biophys. J. 1, 445–466 (1961).
R. Hifer, Application of Fractional Calculus in Physics (World Scientific, Singapore, 2000).
A. L. Hodgkin and A. F. Huxley, “A quantitative description of membrane current and application to conduction and excitation in nerve,” J. Physiol. 117, 500–544 (1954).
A. L. Hodgkin, “The local changes associated with repetitive action in a non-modulated axon,” J. Physiol. 107, 165–181 (1948).
F. C. Hoppensteadt and E. M. Izhikevich, Weakly Connected Neural Networks (Springer, New York, 1997).
E. M. Izhikevich, “Class 1 neural excitability, conventional synapses, weakly confornected networks and mathematical foundations of pulse coupled models,” IEEE Trans. Neural Networks 10, 499–507 (1999).
E. M. Izhikevich, “Neural excitability, spiking and bursting,” Int. J. Bifurcat. Chaos 10, 1171–1266 (2000).
E. M. Izhikevich, “Subcritical elliptic bursting of bautin type,” SIAM J. Appl. Math. 60, 503–535 (2000).
A. A. Kilbas, H. M. Srivastava, and J. J. Trujilo, Theory and Applications of Fractional Differential Equations (Elsevier, Amsterdam, 2006).
P. Kumar and O. P. Agrawal, “An approximate method for numerical solution of fractional differential equations,” Signal Proccess. 6, 2602–2610 (2006).
C. P. Li and Y. H. Wang, “Numerical algorithm based on Adomian decomposition for fractional differential equations,” Comput. Math. Appl. 57, 1627–1681 (2009).
R. L. Magin, “Fractional calculus in bioengineering,” Crit. Rev. Biomed. Eng. 32, 1–104 (2004).
D. Mishra, A. Yadav, and P. K. Kalra, “Chaotic behavior in neural network and FitzHugh-Nagumo neuronal model,” Lect. Notes Comput. Sci. 3316, 868–873 (2004).
C. A. Monje, Y. Q. Chen, B. M. Vinagre, D. Y. Xue, and V. Feliu, Fractional Order Systems and Controls: Fundamentals and Applications (Berlin, Springer, 2010).
C. Morris and H. Lecar, “Voltage oscillations in the Barnacle giant muscle fiber,” J. Biophys. 35, 193–213 (1981).
I. Petras, Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Springer, London, Beijing, 2011).
J. Rinzel, “Models in neurobiology,” in Nonlinear Phenomena in Physics and Biology (Plenum, New York, 1981), pp. 345–367.
J. Rinzel and G. B. Ermentrout, “Analysis of neural excitability and oscillations,” in Methods in Neuronal Modeling (MIT, Cambridge, MA, 1989).
J. Rinzel, “A formal classification of bursting mechanisms in excitable systems,” Lect. Notes Biomath. 71, 267–281 (1987).
M. Shi and Z. Wang, “Abundant bursting patterns of a fractional-order Morris-Lecar neuron model,” Commun. Nonlin. Sci. Numer. Simulat. 19, 1956–1969 (2014).
H. Wang and Q. Wang, “Bursting oscillations, bifurcation and synchronization in neuronal systems,” Chaos, Solitons, Fractals 44, 667–675 (2011).
S. Westerlund and L. Ekstam, “Capacitor theory,” IEEE Trans. Dielectr. Electr. Insul. 1 (8), 26–39 (1994).
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Alidousti, J., Ghaziani, R.K. Spiking and bursting of a fractional order of the modified FitzHugh-Nagumo neuron model. Math Models Comput Simul 9, 390–403 (2017). https://doi.org/10.1134/S2070048217030036
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DOI: https://doi.org/10.1134/S2070048217030036