A martingale analysis of first passage times of time-dependent Wiener diffusion models

Research in psychology and neuroscience has successfully modeled decision making as a process of noisy evidence accumulation to a decision bound. While there are several variants and implementations of this idea, the majority of these models make use of a noisy accumulation between two absorbing boundaries. A common assumption of these models is that decision parameters, e.g., the rate of accumulation (drift rate), remain fixed over the course of a decision, allowing the derivation of analytic formulas for the probabilities of hitting the upper or lower decision threshold, and the mean decision time. There is reason to believe, however, that many types of behavior would be better described by a model in which the parameters were allowed to vary over the course of the decision process. In this paper, we use martingale theory to derive formulas for the mean decision time, hitting probabilities, and first passage time (FPT) densities of a Wiener process with time-varying drift between two time-varying absorbing boundaries. This model was first studied by Ratcliff (1980) in the two-stage form, and here we consider the same model for an arbitrary number of stages (i.e. intervals of time during which parameters are constant). Our calculations enable direct computation of mean decision times and hitting probabilities for the associated multistage process. We also provide a review of how martingale theory may be used to analyze similar models employing Wiener processes by re-deriving some classical results. In concert with a variety of numerical tools already available, the current derivations should encourage mathematical analysis of more complex models of decision making with time-varying evidence

Dynamical analysis of Bayesian inference models for the Eriksen task

The Eriksen task is a classical paradigm that explores the effects of competing sensory inputs on response tendencies, and the nature of selective attention in controlling these processes. In this task, conflicting flanker stimuli interfere with the processing of a central target, especially on short reaction-time trials. This task has been modeled by neural networks and more recently by a normative Bayesian account. Here, we analyze the dynamics of the Bayesian models, which are nonlinear, coupled discrete-time dynamical systems, by considering simplified, approximate systems that are linear and decoupled. Analytical solutions of these allow us to describe how posterior probabilities and psychometric functions depend upon model parameters. We compare our results with numerical simulations of the original models and derive fits to experimental data, showing that agreements are rather good. We also investigate continuum limits of these simplified dynamical systems, and demonstrate that Bayesian updating is closely related to a drift-diffusion process, whose implementation in neural network models has been extensively studied. This provides insight on how neural substrates can implement Bayesian computations