The iris constricts under increased illuminance and dilates when illuminance falls. The response is well modelled as a first-order lag preceded by a transport delay, giving a transfer function with a single pole and an exponential delay term.
Retinal photoreceptors encode the illuminance $L(t)$ and signal the pretectal nucleus. Efferent parasympathetic fibres then drive the sphincter pupillae to contract, reducing the pupil area $A(t)$. The afferent and efferent pathway together introduces a transport delay $T_d$ of approximately $180$ to $250\text{ ms}$ before any movement is observed.
The iris musculature is over-damped and is well approximated as a first-order lag with time constant $\tau \approx 200$ to $300\text{ ms}$. Combining the delay and the lag gives the linear model
where $a(t) = A(t) - A_0$ is the deviation of pupil area from its dark-adapted resting value $A_0$. The negative sign reflects the inverse relationship between illuminance and pupil area. The gain $K$ has units of area per unit illuminance.
Apply the Laplace transform to both sides of the governing ODE under zero initial conditions, $a(0) = 0$. The derivative maps to multiplication by $s$, and the delayed input $L(t - T_d)$ maps to $e^{-T_d s}\,L(s)$ by the time-shift property.
Divide through to obtain the transfer function $G(s) = A(s)/L(s)$:
The expression separates into three factors: a static gain $-K$, a first-order lag $1/(\tau s + 1)$ with a single pole at $s = -1/\tau$, and a transport-delay term $e^{-T_d s}$. The delay term contributes no poles or zeros but adds a phase lag of $\omega T_d$ at each frequency.
For an illuminance step of size $L_0$ applied at $t = 0$, the inverse Laplace transform gives
The output remains zero for $t < T_d$, since the signal has not yet reached the iris. For $t \ge T_d$ the pupil area decays exponentially toward the steady-state value $A_\infty = A_0 - K L_0$, closing approximately $63\%$ of the remaining gap per time constant $\tau$. The pupil in the adjacent panel is driven by this model directly: the ring diameter is recomputed from $a(t)$ at each frame.
The preset row selects alternative input waveforms: square-wave flashing, sinusoidal modulation, a brief single pulse, and a linear ramp. For periodic inputs the steady-state output follows the frequency response $G(j\omega) = -K\,e^{-j\omega T_d}/(1 + j\omega\tau)$. Once the driving frequency exceeds $\omega_c = 1/\tau$, the lag attenuates each cycle and the pupil ceases to track the input faithfully.