For non-corresponding points in the images (i.e.,
x ≠
x′ − Δ
x or
y ≠
y′), the values of
ɛ are uncorrelated, and so the product of the images averages to
μ 2. For corresponding points (i.e., where
x =
x′ − Δ
x and
y =
y′), the values of
ɛ are identical, and so there we pick up an additional term that depends on the variance of
ɛ:
Thus,
and similarly
for 〈
v R 2〉. Using these results, we can write the mean energy-model response as
where
From
Equation A8, we can divide this mean response into a baseline response that would be observed even with binocularly uncorrelated stimuli
and a disparity-modulated term 〈
ɛ 2〉
B(Δ
x). In the uninteresting case where the images are blank, 〈
ɛ 2〉 = 0 and so there is no disparity modulation. Otherwise, the amplitude of the disparity tuning curve relative to the baseline is
where Δ
x pref is defined as the disparity that maximizes the magnitude of the disparity-modulated term.
L, R, M, and Δ
x pref all depend only on the particular receptive field functions, i.e., the properties of the neuronal population encoding disparity. The only term that depends on the image statistics is
μ 2/〈
ɛ 2〉. This term is multiplied by (
L +
R), the integral of the receptive field functions. For the special case of odd-symmetric or very narrow-band cells, this integral is zero. In this case, the amplitude ceases to depend on the image statistics and is simply
A =
B(Δ
pref)/
M. Where the integral (
L +
R) is non-zero, it is clear by inspecting
Equation A11 that
A is maximized when the image has no DC component, i.e.,
μ = 0. Then,
A =
B(Δ
pref)/
M. Any non-zero value of
μ reduces
A, the amplitude of the disparity-modulated response. This is the reason for the difference between the mixed- and same-polarity stimuli in
Figure 11.