Brain-inspired neuromorphic systems have attracted much attention as new computing paradigms for power-efficient computation. Here, we report a silicon synaptic transistor with two electrically independent gates to realize a hardware-based neural network system without any switching components. The spike-timing dependent plasticity characteristics of the synaptic devices are measured and analyzed. With the help of the device model based on the measured data, the pattern recognition capability of the hardware-based spiking neural network systems is demonstrated using the modified national institute of standards and technology handwritten dataset. By comparing systems with and without inhibitory synapse part, it is confirmed that the inhibitory synapse part is an essential element in obtaining effective and high pattern classification capability.