Figure 2.
Illustration of the perceptual and response models. The perceptual model comprises three hierarchical states (x1, x2, and x3). x2 and x3 evolve in time as hierarchically coupled Gaussian random walks (see equations on the left), and x2 determines the probability that the target appears at the cued location (x1 = 1). The session-specific parameters ω and ϑ affect the updating of the beliefs about the states x and are estimated from individual RS data on the basis of the attentional weight α(π̂1) that depends on the precision on the first level of the (inverted) perceptual model. Here, RS is supposed to vary linearly with α(π̂1) in valid trials and with 1 − α(π̂1) in invalid trials (right). Whereas ζ1ν and ζ1i determine the intercepts (i.e., the absolute level of RS) on valid and invalid trials, respectively, ζ2 governs the slope of the linear function and hence the strength of the association between RS and the attentional weight α(π̂1) as derived from the perceptual model. Ellipses represent constants; diamonds represent quantities that change with time (i.e., that carry a time index). Hexagons, like diamonds, represent quantities that change with time but additionally depend on their previous state in time in a Markovian fashion.