Here we are primarily interested in understanding how subjects integrate visual information from the random-dot flow field with nonvisual (such as proprioceptive and vestibular) cues. We use deviations in the center of pressure as a measure of this perceptual cue combination, and the 2AFC task provides uncertainty measurements for the visual cues. Under the standard optimal cue combination model, cues are combined linearly with weights proportional to their relative precision. Assuming that cues are independent and the prior is noninformative, we say that the probability distribution for the combined, visual–nonvisual estimate of posture is given by
where
x is the combined postural estimate,
y vis describes the magnitude of the visual cue, and
y nonvis describes magnitude of the nonvisual cue. We assume that both the visual and nonvisual cues have Gaussian noise about them.
N(
μ, σ) denotes a normal distribution with mean
μ and standard deviation
σ, and the partition function,
Z, ensures that the distribution integrates to 1. We assume that the mean of the visual estimate
μ vis is given by the stimulus, with uncertainty
σ vis from the 2AFC task. For passive standing, the nonvisual estimate is assumed to have mean 0, and
σ nonvis is a free parameter for the uncertainty of the nonvisual cues. The combined estimate of the subject's posture is then given by
This 2AFC-calibrated cue combination model has two free parameters: the uncertainty of the nonvisual cues
σ nonvis and a linear scaling factor that maps perceptual postural estimates
to the observed postural responses. The uncertainty about the visual cues is completely determined by a Weber's law fit to the JNDs estimated during the 2AFC task.
Note that we deliberately avoid describing posture in terms of COP position or velocity. Although a number of studies frame this cue combination problem as a linear feedback control problem where the visual information acts via a multiplicative gain on postural position or velocity, we are primarily interested in the effect of Weber's law scaling of uncertainty. Both position and velocity should be affected by the scaling of visual uncertainty with velocity, and a number of features of the postural response could be used to measure its effect. Here we use the integral of the anterior–posterior COP deviation in a short time window following presentation of the stimulus. We find this feature of the response to be the most stable.