Abstract
In this paper, the dynamical behaviors of a fractional order Hindmarsh-Rose neuronal model are studied. First, based on the stability theory of fractional order systems, some sufficient conditions for the stability and Hpof-type bifurcation are given for such fractional order system. Then, the frequency and amplitude of periodic oscillations are determined by numerical simulations. It is shown that the frequency of oscillations incurs a small variation with respect to different values of the order, while the amplitude of oscillations gets larger as the order is increased. Numerical simulations are performed to verified the theoretical results.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)
Bagley, R.L., Calico, R.A.: Fractional Order State Equations for the Control of Viscoelastically Damped Structures. J. Guid. Control Dyn. 14, 304–311 (1991)
Sun, H.H., Abdelwahad, A.A., Onaral, B.: Linear Approximation of Transfer Function with a Pole of Fractional Order. IEEE Trans Autom. Control AC 29, 441–444 (1984)
Ichise, M., Nagayanagi, Y., Kojima, T.: An Analog Simulation of Noninteger Order Transfer Functions for Analysis of Electrode Process. J. Electroanal. Chem. 33, 253–265 (1971)
Heaviside, O.: Electromagnetic Theory. Chelsea, New York (1971)
Laskin, N.: Fractional Market Dynamics. Phys. A 287, 482–492 (2000)
Kusnezov, D., Bulgac, A., Dang, G.D.: Quantum Levy Processes and Fractional Kinetics. Phys. Rev. Lett. 82, 1136–1139 (1999)
Cole, K.S.: Electric Conductance of Biological Systems. In: Proc. Cold Spring Harbor Symp. Quant. Biol., New York, pp. 107–116 (1993)
Anastasio, T.J.: The Fractional Order Dynamics of Brainstem Vestibuleoculumotor Neurons. Biol. Cybern. 72, 69–79 (1994)
Gopalsamy, K., Leung, I.: Convergence under Dynamical Thresholds with Delays. IEEE Trans. Neural Netw. 8, 341–348 (1997)
Xu, X., Hua, H.Y., Wang, H.L.: Stability Switches, Hopf Bifurcation and Chaos of a Neuron Model with Delay-Dependent Parameters. Phys. Lett. A 354, 126–136 (2006)
Cao, J., Xiao, M.: Stability and Hopf bifurcation in a Simplified BAM Neural Network with Two Time Delays. IEEE Trans. Neural Netw. 18, 416–430 (2007)
Matignon, D.: Stability Results for Fractional Differential Equations with Applications to Control Processing. In: Proceedings IMACS-SMC 1996, Lille, France, pp. 963–968 (1996)
Ahmed, E., El-Sayed, A., El-Saka, H.: Equilibrium Points, Stability and Numerical Solutions of Fractional Order Predator-Prey and Rabies Models. J. Math. Anal. Appl. 325, 542–553 (2007)
FitzHugh, R.: Impulses and Physiological State in Theoretical Models of Nerve Membrane. Biophy. J. 1, 445–467 (1961)
Nagumo, J., Arimoto, S., Yoshizawa, S.: An Active Pulse Transmission Line Simulating Nerve Axon. In: Proc. IRE, vol. 50, pp. 2061–2070 (1962)
Tsuji, S., Ueta, T., Kamakami, H., Fujii, H., Aihara, K.: Bifurcations in Two-Dimensional Hindmarsh-Rose Type Model. Int. J. Bifurc. Chaos 17, 985–998 (2007)
Gantmacher, F.R.: The Theory of Matrices. Chelsea, New York (1959)
Diethelm, K., Ford, N.J., Freed, A.D.: A predictor-corrector approach for the numerical solution of fractional differential equations. Nonlin. Dynam. 29, 3–22 (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Xiao, M. (2012). Stability Analysis and Hopf-Type Bifurcation of a Fractional Order Hindmarsh-Rose Neuronal Model. In: Wang, J., Yen, G.G., Polycarpou, M.M. (eds) Advances in Neural Networks – ISNN 2012. ISNN 2012. Lecture Notes in Computer Science, vol 7367. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31346-2_25
Download citation
DOI: https://doi.org/10.1007/978-3-642-31346-2_25
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31345-5
Online ISBN: 978-3-642-31346-2
eBook Packages: Computer ScienceComputer Science (R0)