Supplementary material A
Feedback effect analyses
The measurements were recorded under closed-loop conditions and so we determined whether there was a feedback effect of the sinusoidal modulation on pupil size. The physiological feedback between the retina and pupil that changes the pupil area in order to alter retinal illumination has been studied from a bioengineering point of view as an example of a biological control system.
1–3 In this system, the signal at the control mechanism is compared with a reference signal in order to maintain a stable output signal. This comparison produces a feedback signal from the output that modifies the retinal illumination through the change of the pupil size. In the control system, this feedback is referred to as closed loop, while open loop represents the behavior of the system when there is no feedback. An open loop system can be accomplished by measuring pupil size under a consensual condition or in Maxwellian view.
4,5 We compared the outputs of these open and closed loop models for our stimulus conditions to assess whether feedback was significant.
We analyzed the role of the physiologic feedback effect on the amplitude and phase response (Fig. 1) by computing the transfer functions in open loop (without feedback) and closed loop (with feedback) with a time delay and a third-order lag derived by Sherman and Stark.
6 Although other non-linear properties of the pupil response could be accounted for in a more general model, 2,3 the Sherman and Stark

linear approach is sufficient to model our data. The transfer function implemented was expressed as a function of complex frequency operator s (Eq. A1).
𝐹(𝑠) = 𝑏
𝐺(𝑠)
1+𝐻𝐺(𝑠)
(A1) where b was a constant of proportionality fixed arbitrarily at 0.008, H represented the feedback attenuation that was computed as the relation of retinal illuminantion change due to the pupil variation (0.53%; see below) divided the contrast of the stimulation(25%), and G was the transfer function without feedback (Eq. A2).
𝐺(𝑠) = 𝑔 𝑒
−𝜏𝑠
(1+ 𝑠 𝜔0
)(1+ 𝑠 𝜔1
)(1+ 𝑠 𝜔2
)
(A2)
Whit the parameters optimized to g=2, τ=0.215, ω
0
=18.2, ω
1
=10 and ω
2
=12.5. Both open loop and closed loop transfer function curves for the isolated rod stimuli at 0 log cd/m 2 are shown in Fig. 1A. Clearly, both the open and closed loop estimates of the amplitude and phase data are similar at <= 2Hz, but at 4Hz and 8Hz, the model predictions of phase diverge. The simulation in the open loop condition was consistent with previous results.
7 For the three observers at 1Hz, the pupil response amplitude for the combined rod and cone excitation was on average a 0.11 mm change at -0.9 log cd/m 2 relative to the 4.17 mm baseline pupil diameter on the steady adapting background, and a 0.16 mm change at 0 log cd/m 2 relative to the
3.43mm baseline pupil diameter. In other words, the amplitude change was equivalent to 0.07% (-0.9 log cd/m 2 ) and 0.22% (0 log cd/m 2 ) relative to the baseline pupil size, which results in negligible changes in retinal illuminance (less than 0.53%). To further confirm the insignificant effect of the feedback at 1Hz, one observer completed measurements in consensual conditions (open loop) and

binocular conditions (closed loop) with isolated rod stimuli, cone stimuli and combined rod and cone stimuli at four phase differences (0deg, 90deg, 180deg and
270deg) between the rod and cone stimulations at 0 log cd/m 2 (Fig. 1B). The consensual measurement was carried out after dilating the right eye (tropicamide
1%). While the stimulation was presented to the right eye, the pupil diameter of the left eye in dark conditions was measured. Fig. 1B shows that the amplitudes of the consensual measurement were slightly lower (18% in average) than binocular condition, consistent with the literature.
8 Importantly, both the consensual and binocular measurements had similar phases, indicating that the feedback was minimal at 1Hz and therefore this frequency is appropriate to examine the amplitude and phase relationships (and latencies) between the inner and outer retina photoreceptor inputs to the pupil control pathway in Experiment 2.
A B
Fig. 1. A) Pupil responses for “Rod stimuli” condition of experiment 1 at 0 log cd/m 2 .
The fits of the transfer function in open loop and closed loop are also plotted in the figure (see Appendix A for the functions and parameters values). B) Consensual response compared with binocular response for observer PB.

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