The Pupillary Light Reflex as a Dynamical Arousal Index: A Delay-Differential Framework for Non-Invasive Locus Coeruleus Estimation

This paper argues that the re-dilation trajectory of the Pupillary Light Reflex, which standard pupillometry discards, encodes the loop gain Gof a nonlinear delayed negative-feedback system whose operating point is set by tonic Locus Coeruleus output. The central theoretical contribution is an exact analytical solution for the recovery time constant τreturn(G) derived via the Lambert W function, which eliminates the classical zero-delay approximation and correctly locates the Hopf bifurcation at Gc. Full nonlinear simulations of the Longtin-Milton model validate the calibration curve, and a spectral cross-validation independently confirms the bifurcation geometry. From this framework a three-variable instrument architecture which combines resting diameter, recovery dynamics, and a dual-wavelength chromatic post-illumination response,is derived. The work is theoretical and computational throughout; the instrument specification is a principled design consequence of the validated framework, not a clinical prototype, and three falsifiable predictions define the bridge to empirical work.