In this work, we investigate organic electrochemical transistors (OECTs) as a novel artificial electronic device for the realization of synaptic behavior, bioelectronics, and a variety of applications. A numerical method considering the Poisson-Boltzmann statistics is introduced to reproduce associated charge densities, electrostatics and switching properties of OECTs. We shed light on the working principle of OECTs by taking into account the ionic charge distribution in the electrolyte and incomplete ionization of the organic semiconductor describing the underlying electrochemical redox reaction. This enables analyzing the OECTs electrical performance as well as a simplified chemical properties via an electrical double layer, doping and de-doping of the OMIEC layer. We have fabricated, characterized, simulated and analyzed OECTs based on PEDOT:PSS, and we show that the proposed model reveals important properties of the device's working mechanism. The model shows a good agreement with the experimental data of the fabricated devices.