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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 836402, 12 pages
http://dx.doi.org/10.1155/2015/836402
Research Article

Qualitative Analysis of a Quadratic Integrate-and-Fire Neuron Model with State-Dependent Feedback Control

1Key Laboratory of Biologic Resources Protection and Utilization, Hubei Minzu University, Enshi, Hubei 445000, China
2Department of Mathematics, Chongqing Jiaotong University, Chongqing 400074, China
3College of Mathematics and Information Science, Shaanxi Normal University, Xi’an 710062, China

Received 7 May 2015; Revised 14 August 2015; Accepted 30 August 2015

Academic Editor: Michael Radin

Copyright © 2015 Guangyao Tang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Spiking neuron models which exhibit rich dynamics are usually defined by hybrid dynamical systems. It is revealed that mathematical analysis of these models has important significance. Therefore, in this work, we provide a comprehensively qualitative analysis for a quadratic integrate-and-fire model by using the theories of hybrid dynamical system. Firstly, the exact impulsive and phase sets are defined according to the phase portraits of the proposed model, and then the Poincaré map is constructed. Furthermore, the conditions for the existence and stability of an order 1 periodic solution are provided. Moreover, the existence and nonexistence of an order periodic solution have been studied theoretically and numerically, and the results show that the system has periodic solutions with any period. Finally, some biological implications of the mathematical results are discussed.