Diffusion theory of the antipodal “shadow” mode in continuous-outcome, coherent-motion decisions.

Abstract

Continuous-outcome decisions, in which responses are made on continuous scales, are increasingly used to study perception and memory for stimulus attributes like color, orientation, and motion. This interest has led to the development of models of continuous-outcome decision processes like the circular diffusion model that predict joint distributions of decision outcomes and response times (RTs). We use the circular diffusion model and a new spherical generalization of it to model performance in a continuous-outcome version of the random-dot motion task. The task is a benchmark test of decision models because it yields bimodal distributions of decision outcomes: In addition to a peak or mode in the true direction of motion, there is a secondary, antipodal, mode at 180° to the true direction. Models like the circular diffusion model, in which evidence is accumulated by a single process, are thought to be unable to predict bimodality. We compared diffusion models for the continuous motion task in which evidence is accumulated in either a two-dimensional (2D) or a three-dimensional (3D) representational space. We found that performance was well described by a spherical (3D) diffusion model in which the drift rate encodes perceived motion direction and strength and the points on the bounding sphere representing the decision criterion are projected onto a 2D circle to make a response. A model with an antipodal component of drift rate and drift-rate variability successfully predicted bimodal distributions of decision outcomes and the joint distributions of decision outcomes and RT for individual participants.

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

“…When there is across-trial variability in drift rate, Equation 4 is replaced by another expression, whose form depends on whether the components of the distribution of drift rate are independent or correlated. When the components are independent, with means ν 1 and ν 2 and standard deviations η 1 and η 2 , the change-of-measure function takes the form (Smith & Corbett, 2019;Smith et al, 2020Smith et al, , 2022. As in Equation 4, X T denotes the locus of hitting points on the criterion circle.…”

“…It has since become the method of choice for many VWM researchers (Adam et al, 2017;Bays et al, 2009;Oberauer & Lin, 2017;van den Berg et al, 2014;Zhang & Luck, 2008) because it yields entire distributions of retrieval errors rather than just error rates. It has also been used, in related settings, to study perception of, or memory for, color (e.g., Bae, et al, 2015;Hardman et al, 2017;Persaud & Hemmer, 2016), the perception of motion direction (Bae & Luck, 2019Smith et al, 2022), and associative recall from episodic memory (Harlow & Donaldson, 2013;Zhou et al, 2021).…”

“…The development of models that can characterize the decision process at the level of these distributions has consequently become a question of considerable theoretical interest. To date, there have been three models of continuous-outcome decisions: the circular diffusion model (CDM) of Smith and colleagues (Smith, 2016;Smith et al, 2020) and its spherical generalization (Smith et al 2022); the spatially continuous diffusion model (SCDM) of Ratcliff (2018); and the multiply anchored accumulation model Smith 3 (MAAT) of Kvam et al (2022). These models generalize, respectively, the single-process Wiener diffusion model of Ratcliff (1978), the racing, parallel diffusion models of Ratcliff and Smith, (2004), Usher & McClelland (2001), and others, and the linear ballistic accumulator of Brown and Heathcote (2008).…”

“…Consequently, it is not ameliorated by increasing the number of terms in the series representation of the density of the Bessel process. In our published work (Smith, 2016;Smith et al, 2020Smith et al, , 2022Smith & Corbett, 2019;Zhou et al, 2021) we have typically used a series of 50 terms, which more than suffices to yield an accurate representation of the density function in most parts of the parameter space. However, in the parts of the space in which the numerical artifact appears, increasing the number of terms to 500 or 5000 does not diminish its magnitude.…”

“…We show by example that the new method provides a simple and practical way of stabilizing the predictions of the model in those parts of the parameter space that are of interest in fitting data. The method applies to higher-dimensional generalizations of the model, specifically, to the spherical diffusion model (Smith et al, 2022) and the hyperspherical diffusion model (Smith & Corbett, 2019).…”