We start by modeling our data solely in terms of shortening the integration time. We take the now classic approach of implementing the changes in integration time by shortening or lengthening the time constants (
τ) of one or more (
n) cascaded leaky integrating stages (or buffered RC circuits or low-pass filters). The formula for the amplitude response,
A(
f), of
n cascaded leaky integrators is:
and the phase response,
P(
f), is:
where
f is frequency in cycles per second (Hz), and
τ is the time constant in seconds. For further information, see, for example, Watson (
1986). This approach is still relevant in the modern context of molecular processes in the sense that leaky integrators can be associated with first-order biochemical reactions.
The left panels of
Figure 7 show the changes in contrast sensitivity plotted as changes in amplitude, and the right panels show the changes in phase delay. Data for AS, LTS and TC are shown in the top, middle and bottom panels, respectively.
In optimizing the model parameters, the time constants of the
n filters were varied together, thus altering the threshold amplitudes according to
Equation 2 and the phase delays according to
Equation 3. Allowing the time constants of each filter to vary independently yielded additional parameters, but did not significantly improve the predictions. The phase and amplitude data were weighted so that their influence was approximately equal (otherwise one or other set of data would dominate the fitting procedure). The number of filters,
n, was allowed to take on non-integer values in preliminary fits. A value of
n = 2 was used for the final fits, because this represented the best estimate across all three subjects. We emphasize, however, that
n is poorly constrained by the fit, because increases in
n can be offset by decreases in
τ and vice versa. Thus,
n = 2 should be considered approximate, because values of
n > 2 would also produce plausible fits. In contrast, a value of
n = 1 is implausible, because it would limit the maximum phase change to 90°, which is less than the measured changes (see
Figures 4,
5, and
6). The model fits for
n = 2 are shown by the continuous lines in
Figure 7 coded using the same colors as the symbols.
Given that with
n fixed at 2 there is only a single intensity-dependent parameter,
τ, the fits to the phase and amplitude data shown in
Figure 7 are fairly good. (The change in
τ with retinal illuminance for each subject is shown in the lower panel of
Figure 9, below.) Relative to the null model that there is no change in amplitude or phase between levels (i.e., all the values in
Figure 7 are zero and therefore there is effectively no adaptation), the single-parameter model accounts for 97.97% of the threshold amplitude and 89.20% of the phase variance for AS, 96.37% of the threshold amplitude and 75.33% of the phase variance for LTS; and 99.52% of the threshold amplitude and 61.99% of the phase variance for TC. The amplitude fits are generally good, but the phase fits are poorer. Nonetheless, the single parameter model accounts for most of the variance. For TC, the deviations of the phase delays from the model predictions have no obvious structure, and reflect in part the difficulties he had in setting binocular phase delays. For LTS, there is a clear discrepancy in the phase predictions between the two highest levels, where the model seriously underestimates the phase advance. We speculate that this discrepancy is associated with the intrusion of the faster rod pathway (e.g., Sharpe & Stockman,
1999), which seems to become prominent at a lower level for LTS than for the other subjects. We account for this faster pathway by adding an additional time advance of 9.06 ms between the two highest levels; the model then accounts for 98.78% of the amplitude and 91.86% of the phase variance. This speculative modification to the model predictions is shown by the dashed lines for LTS in
Figure 7.