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Spiking neuron models which exhibit rich dynamics are usuallydefined by hybrid dynamical systems. It is revealed thatmathematical analysis of these models has important significance. Therefore, in this work, we provide a comprehensively qualitativeanalysis for a quadratic integrate-and-fire model by using thetheories of hybrid dynamical system. Firstly, the exactimpulsive and phase sets are defined according to the phaseportraits of the proposed model, and then thePoincaré map is constructed. Furthermore, theconditions for the existence and stability of an order 1 periodicsolution are provided. Moreover, the existence and nonexistenceof an order k  k≥2 periodic solution have been studiedtheoretically and numerically, and the results showthat the system has periodic solutions with any period. Finally,some biological implications of the mathematical results arediscussed

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