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

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 Poincare 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 k (k >= 2) 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.