Experimental data indicate that simple motor decisions in vertebrates are preceded by integration of evidence in certain cortical areas, and that the competition between them is resolved in the basal ganglia. While the occurrence of cortical integration is well established, it is not yet clear exactly how the integration occurs. Several models have been proposed, including the race model, the feed forward inhibition (FFI) model and the leaky competing accumulator (LCA) model. In this paper we establish qualitative and quantitative differences between the above mentioned models, with respect to how they are able to initiate the integration process without integrating noise prior to stimulus onset, as well as the models' ability to terminate the integration after a decision has been made, to ensure the possibility of subsequent decisions. Our results show that the LCA model has advantages over the race model and the FFI model in both respects, leading to shorter decision times and an effective termination process.