Optimal control for fast, accurate threshold-hitting

A convenient and popular approach to modeling events in physical processes is to equip a latent dynamical model with a threshold that, when hit, indicates occurrence. In a multidimensional construct, each latent state has its own individual threshold corresponding to independent events. The overall threshold boundary can be understood as a hypercube in the latent space, with each edge corresponding to a different event. In this scenario, it is of interest to understand the optimal driving inputs (of the latent states) that can induce robust events quickly and accurately, i.e., where thresholds are hit near the center of the faces of the hypercube, away from competing thresholds. Here we study a binary version of this optimal threshold-hitting control problem, relevant to questions in theoretical neuroscience. We fully characterize the optimal solution from a geometric standpoint and show that the speed-accuracy trade-off in these problems drives an inhibitory input mechanism that may further result in paradoxical nonexistence of an optimal input, resembling classical problems from calculus of variations.