Appendix A. Mathematical details of decision optimization in two-alternative forced choice tasks

In this appendix we collect and describe mathematical results for first passage, boundaryless, and reflecting-boundary drift diffusion problems. We consider both diffusion and OrnsteinUhlenbeck (O-U) models. The main tools are drawn from applied probability, the theory of stochastic ordinary differential equations, and classical perturbation and asymptotic methods. Here we focus on continuous models in the form of stochastic ordinary differential equations, although we first describe how these arise as limits of the discrete Neyman-Pearson and sequential probability statistical tests, which are the optimal methods of deciding between two alternatives on the basis of noisy accumulating data. It turns out that the optimal continuous processes admit rather simple exact formulae for such behavioral observables as mean decision times (DT) and error rates (ER). Section A.1 reviews the optimality of the sequential probability ratio test, and describes how it becomes the drift diffusion model in the continuum limit. Sections A.2 considers the free response protocol, in which participants make choices in their own time, and Section A.3 addresses the interrogation protocol, in which a decision is deferred until a cue is presented. Section A.4 gives results for extended models in which drift rates and initial conditions are variable.