In this study, we investigate a proton-based three-terminal (3-T) synapse device to realize linear weight-update and I-V linearity characteristics for neuromorphic systems. The conductance states of the 3-T synapse device can be controlled by modulating the proton concentration in the WOx channel. Therefore, we estimate the dynamic change of proton concentration in the channel region, which directly affects synaptic behaviors. Our findings indicate that the supply of an excess number of protons from the SiO2-H electrolyte and low proton diffusivity in the WOx channel result in asymmetric and non-linear weight-update characteristics. In addition, though the linear I-V characteristics can be obtained using non-stoichiometric WOx, we observe that significant oxygen deficiency in the channel region increases the operating current levels. Thus, based on this information, we introduce optimized conditions of each component in the 3-T synapse device and shape of the gate voltage pulses. As a result, an excellent classification accuracy is achieved using linear weight-update and I-V linearity characteristics under optimized device and pulse conditions.