Dynamical Analysis of Bayesian Inference Models for the Eriksen Task

Abstract

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.

Dynamics of attentional selection under conflict: Toward a rational Bayesian account.

“…Elsewhere (Liu et al, 2008), we show that under certain approximating assumptions, the key to the “dip” is that the ratio of the means, a 1 /a 2 , must be within a certain range bounded by functions of the noise variances σ 1 and σ 2 . Intuitively, when a 1 /a 2 is too large, then there is little spatial uncertainty; when a 1 /a 2 is too small, then the inputs lose their spatial specificity.…”

“…Elsewhere (Liu, Yu, & Holmes, 2008), we have shown that the compatibility bias model can be expected to produce a “dip” in the marginal posterior after one sample ( P ( s 2 | X 1 )) under rather loose constraints on the model parameter: β > 3/4 (or more generally, for n flankers, β > ( n + 1)/2 n ). Presumably a dip in the posterior underlies any dip in the decision accuracy for short-RT trials.…”

Perceptual Decisions Formed by Accumulation of Audiovisual Evidence in Prefrontal Cortex

“…Since both semantically congruent and incongruent signals were presented spatiotemporally coincident, subjects may a priori expect auditory and visual signals to emanate from a common source and hence attempt to integrate the signals regardless of their semantic congruency. Indeed, in support of this compatibility bias hypothesis (Liu et al, 2009;Yu et al, 2009), the decrement in performance accuracy for incongruent trials was particularly pronounced at the onset of a trial that was dominated by subjects' congruency prior; it diminished over the course of a trial when subjects accumulated evidence about the true relationship of the signals. Similarly, as indicated by the response time profile across conditions, the effect of an "inappropriate" congruency prior depended on sensory reliability: the response time difference for incongruent relative to congruent trials was particularly pronounced for degraded visual and intact auditory information.…”

“…The interference of incongruent auditory information on the accumulation of visual object evidence should then be particularly pronounced at trial onset when subjects' congruency prior dominates and decreases during the course of the trial, when incoming evidence overrides these prior expectations. The accumulation process is terminated when the evidence about the visual object category reaches a decisional threshold and the subject "opts for" one of the two alternatives (i.e., tool vs musical instrument) [for further details on the implementation of the model, see supplemental material (available at www.jneurosci.org) and Yu et al (2009); for relationship to drift diffusion model, see Liu et al (2009)].…”

Self-Associations Influence Task-Performance through Bayesian Inference

“…Recently Yu et al (2009) have described a normative model of the Eriksen Flanker task, and in later work (Liu et al, 2009) they also provide a connection to DDMs from which they derive semi-analytic formulae for reaction times and error rates. We note that a similar approach is possible for the bivalent rule-association task where, mathematically, the rule cue rather than the flankers act as the “context” variable.…”