The spiking neural network (SNN) is regarded as the third generation of an artificial neural network (ANN). In order to realize a high-performance SNN, an integrate-and-fire (I&F) neuron, one of the key elements in an SNN, must retain the overflow in its membrane after firing. This paper presents an analog CMOS I&F neuron circuit for overflow retaining. Compared with the conventional I&F neuron circuit, the basic operation of the proposed circuit is confirmed in a circuit-level simulation. Furthermore, a single-layer SNN simulation was also performed to demonstrate the effect of the proposed circuit on neural network applications by comparing the raster plots from the circuit-level simulation with those from a high-level simulation. These results demonstrate the potential of the I&F neuron circuit with overflow retaining characteristics to be utilized in upcoming high-performance hardware SNN systems.