Vision for the blind: visual psychophysics and blinded inference for decision models

Evidence accumulation models like the diffusion model are increasingly used by researchers to identify the contributions of sensory and decisional factors to the speed and accuracy of decision-making. Drift rates, decision criteria, and nondecision times estimated from such models provide meaningful estimates of the quality of evidence in the stimulus, the bias and caution in the decision process, and the duration of nondecision processes. Recently, Dutilh et al. (Psychonomic Bulletin & Review 26, 1051–1069, 2019) carried out a large-scale, blinded validation study of decision models using the random dot motion (RDM) task. They found that the parameters of the diffusion model were generally well recovered, but there was a pervasive failure of selective influence, such that manipulations of evidence quality, decision bias, and caution also affected estimated nondecision times. This failure casts doubt on the psychometric validity of such estimates. Here we argue that the RDM task has unusual perceptual characteristics that may be better described by a model in which drift and diffusion rates increase over time rather than turn on abruptly. We reanalyze the Dutilh et al. data using models with abrupt and continuous-onset drift and diffusion rates and find that the continuous-onset model provides a better overall fit and more meaningful parameter estimates, which accord with the known psychophysical properties of the RDM task. We argue that further selective influence studies that fail to take into account the visual properties of the evidence entering the decision process are likely to be unproductive.

A Numerical Schemefor the Probability Density of the First Hitting Time for Some Random Processes

Departing from a general stochastic model for a moving boundary problem, we consider the density function of probability for the first passing time. It is well known that the distribution of this random variable satisfies a problem ruled by an advection–diffusion system for which very few solutions are known in exact form. The model considers also a deterministic source, and the coefficients of this equation are functions with sufficient regularity. A numerical scheme is designed to estimate the solutions of the initial-boundary-value problem. We prove rigorously that the numerical model is capable of preserving the main characteristics of the solutions of the stochastic model, that is, positivity, boundedness and monotonicity. The scheme has spatial symmetry, and it is theoretically analyzed for consistency, stability and convergence. Some numerical simulations are carried out in this work to assess the capability of the discrete model to preserve the main structural features of the solutions of the model. Moreover, a numerical study confirms the efficiency of the scheme, in agreement with the mathematical results obtained in this work.

Receiver operating characteristic curves for a simple stochastic process that carries a static signal

The detection of a weak signal in the presence of noise is an important problem that is often studied in terms of the receiver operating characteristic (ROC) curve, in which the probability of correct detection is plotted against the probability for a false positive. This kind of analysis is typically applied to the situation in which signal and noise are stochastic variables; the detection problem emerges, however, also often in a context in which both signal and noise are stochastic processes and the (correct or false) detection is said to take place when the process crosses a threshold in a given time window. Here we consider the problem for a combination of a static signal which has to be detected against a dynamic noise process, the well-known Ornstein-Uhlenbeck process. We give exact (but difficult to evaluate) quadrature expressions for the detection rates for false positives and correct detections, investigate systematically a simple sampling approximation suggested earlier, compare to an approximation by Stratonovich for the limit of high threshold, and briefly explore the case of multiplicative signal; all theoretical results are compared to extensive numerical simulations of the corresponding Langevin equation. Our results demonstrate that the sampling approximation provides a reasonable description of the ROC curve for this system, and it clarifies limit cases for the ROC curve.

First-passage-time densities for time-non-homogeneous diffusion processes

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].

First-passage-time densities for time-non-homogeneous diffusion processes

In this paper we study a Volterra integral equation of the second kind, including two arbitrary continuous functions, in order to determine first-passage-time probability density functions through time-dependent boundaries for time-non-homogeneous one-dimensional diffusion processes with natural boundaries. These results generalize those which were obtained for time-homogeneous diffusion processes by Giorno et al. [3], and for some particular classes of time-non-homogeneous diffusion processes by Gutiérrez et al. [4], [5].

Optimal estimation of diffusion processes hidden by general obstacles

Let Xt be an n-dimensional diffusion process and S(t) be a set-valued function. Suppose Xt is invisible when it is hidden by S(t), but we can see the process exactly otherwise. In this paper, we derive the optimal estimator E[f(X1) | Xs1Xs∉S(s), 0 ≤ s ≤ 1] for a bounded Borel function f. We illustrate some computations for Gauss-Markov processes.

A simple algorithm to generate firing times for leaky integrate-and-fire neuronal model

A method to generate first passage times for a class of stochastic processes is proposed. It does not require construction of the trajectories as usually needed in simulation studies, but is based on an integral equation whose unknown quantity is the probability density function of the studied first passage times and on the application of the hazard rate method. The proposed procedure is particularly efficient in the case of the Ornstein-Uhlenbeck process, which is important for modeling spiking neuronal activity.

Effect of an exponentially decaying threshold on the firing statistics of a stochastic integrate-and-fire neuron

We study a white-noise driven integrate-and-fire (IF) neuron with a time-dependent threshold. We give analytical expressions for mean and variance of the interspike interval assuming that the modification of the threshold value is small. It is shown that the variability of the interval can become both smaller or larger than in the case of constant threshold depending on the decay rate of threshold. We also show that the relative variability is minimal for a certain finite decay rate of the threshold. Furthermore, for slow threshold decay the leaky IF model shows a minimum in the coefficient of variation whenever the firing rate of the neuron matches the decay rate of the threshold. This novel effect can be seen if the firing rate is changed by varying the noise intensity or the mean input current.

Stochastic Dynamic Models of Response Time and Accuracy: A Foundational Primer

A large class of statistical decision models for performance in simple information processing tasks can be described by linear, first-order, stochastic differential equations (SDEs), whose solutions are diffusion processes. In such models, the first passage time for the diffusion process through a response criterion determines the time at which an observer makes a decision about the identity of a stimulus. Because the assumptions of many cognitive models lead to SDEs that are time inhomogeneous, classical methods for solving such first passage time problems are usually inapplicable. In contrast, recent integral equation methods often yield solutions to both the one-sided and the two-sided first passage time problems, even in the presence of time inhomogeneity. These methods, which are of particular relevance to the cognitive modeler, are described in detail, together with illustrative applications. Copyright 2000 Academic Press.