Dynamical Behavior Analysis of a Neutral-Type Single Neuron System

In this paper, the equation used to describe a single neuron system of neutral-type with one delay is studied. By using a suitable Lyapunov function, we derive sufficient conditions of globally asymptotic stability of the zero equilibrium point of this equation. Moreover, the criterions of the existence of Hopf bifurcation that gives rise to self-sustained oscillation are obtained. Finally, the perturbation theory is used to acquire the expansion of oscillation to any order on the basis of the coefficient of the fundamental frequency, and the frequency-amplitude relations about the second order are carefully discussed.

[1]  Qingkai Kong,et al.  Oscillation Theory for Functional Di erential Equations , 1994 .

[2]  X. Liao,et al.  Dynamics of the congestion control model in underwater wireless sensor networks with time delay , 2016 .

[3]  Aleksandra Badura,et al.  The Cerebellum, Sensitive Periods, and Autism , 2014, Neuron.

[4]  M. J. Field,et al.  Heteroclinic Networks in Homogeneous and Heterogeneous Identical Cell Systems , 2015, Journal of Nonlinear Science.

[5]  Xiaofeng Liao,et al.  Dynamical Behavior of Chua's Circuit With Lossless Transmission Line , 2016, IEEE Transactions on Circuits and Systems I: Regular Papers.

[6]  Amir F. Atiya,et al.  How delays affect neural dynamics and learning , 1994, IEEE Trans. Neural Networks.

[7]  K. Gopalsamy.  Stability and Oscillations in Delay Differential Equations of Population Dynamics , 1992 .

[8]  A. Hatley.  Mathematics in Science and Engineering , Volume 6: Differential- Difference Equations. Richard Bellman and Kenneth L. Cooke. Academic Press, New York and London. 462 pp. 114s. 6d. , 1963, The Journal of the Royal Aeronautical Society.

[9]  J J Hopfield,et al.  Neurons with graded response have collective computational properties like those of two-state neurons. , 1984, Proceedings of the National Academy of Sciences of the United States of America.

[10]  G. Ladas,et al.  Oscillation Theory of Delay Differential Equations: With Applications , 1992 .

[11]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[12]  Guanrong Chen,et al.  Novel robust stability criteria for interval-delayed Hopfield neural networks , 2001 .