Models based on statistical decision theory
9,10,15,17 have assumed that visual performance is limited by various sources of noise. One source is an additive internal noise, due to the variability of neural firing rate, which does not depend on the level of external stimulation. Other sources are multiplicative internal noises,
18 which are proportional to the energy of the signal or the density of the external noise. In addition, human performance depends on the sampling efficiency, which refers to the observer's ability to match an internal template to the signal profile or to integrate signals over the entire signal area.
Lu and Dosher
19 proposed a perceptual template model for the detection of luminance patterns (
Fig. 2) which consists of a filter (the perceptual template) a nonlinear transducer, multiplicative internal noises with magnitudes that are monotonic functions of the energy of the signal and the external noise, an additive internal noise, and a decision process.
According to this model, the detectability index (
d′, determined by the signal-to-noise ratio at the decision stage) for a signal embedded in external noise can be expressed as
where
E is the contrast energy of the signal (the integral of the squared luminance function),
k is the efficiency with which humans use the perceptual template to match the signal,
N is the density of the external noise, γ is the exponent of the power transducer function,
N add is the additive internal noise with amplitude that does not depend on the input, and
m and
s are coefficients that determine the equivalent multiplicative noises induced by the external noise (
mN γ) and the signal (
s(
kE)
γ), respectively. In this version of the perceptual template model, we used different multiplicative coefficients (
m and
s) to separate the effects of external noise and signal on the induced multiplicative internal noise. The sensitivities of these two noise components was considered independently, since the component due to the external noise reflects the variability of pooled activity of channels tuned to a wide range of spatial frequencies and orientations, whereas the component due to the signal represents the variability of activity of a spatial filter matched to the test disk.
The threshold contrast [
C = (
E/
E 0)
1/2] can be represented by rearrangement of
equation 1 where
E 0 represents the energy of the signal with unit contrast (
E 0 = 45,600 μdeg
2).
Equation 2 predicts that the threshold contrast necessary to detect the target depends on the variance of the external noise added to the stimulus. The solid line in
Figure 3 shows the typical behavior of a threshold contrast/external noise variance function: Thresholds remain constant when small amounts of external noise are added up to a certain level and rise when contrasts are increased beyond that level.
If migraineurs had suboptimal sampling efficiency (reduced
k) for the detection of luminance patterns, they would require more contrast, shifting the entire threshold/noise variance function upward (
Fig. 3, dotted line). The same upward shift would be predicted if the target (signal) induced abnormally high levels of internal noise (increased
s). However, if migraineurs experienced increased additive internal noise with an amplitude that is independent of the external stimulation (
N add), greater contrast would be needed for detection only at low but not at high levels of external noise (
Fig. 3, dotted-dashed line). Finally, if the limitation were linked to the amount of internal noise induced by the external noise (increased
m), migraineurs would detect the target equally as well as normal subjects in the absence of external noise but would show deficits when external noise is high (
Fig. 3, dashed line).