Obtaining Stable Predicted Distributions of Response Times and Decision Outcomes for the Circular Diffusion Model

Abstract

The circular diffusion model represents continuous outcome decision making as evidence accumulation by a two-dimensional Wiener process with drift on the interior of a disk, whose radius represents the decision criterion for the task. The hitting point on the circumference of the disk represents the decision outcome and the hitting time represents the decision time. The Girsanov change-of-measure theorem applied to the first-passage time distribution for the Euclidean distance Bessel process yields an explicit expression for the joint distribution of decision outcomes and decision times for the model. A problem with the expression for the joint distribution obtained in this way is that the change-of-measure calculation magnifies numerical noise in the series expression for the Bessel process, which can make the expression unstable at small times when the drift rate or decision criterion is large. We introduce a new method that uses an asymptotic approximation to characterize the Bessel process at short times and the series expression for the large times. The resulting expressions are stable across all parts of the parameter space likely to be of interest in experiments, which greatly simplifies the task of fitting the model to data. The new method applies to the spherical and hyperspherical generalizations of the model and to versions of it in which the drift rates are normally distributed across trials with independent or correlated components.