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

Abstract

Standard pupillometry measures resting diameter and discards everything else. The recovery trajectory following a light pulse, however, encodes the loop gain of the underlying delayed negative feedback system, and that gain is set by tonic Locus Coeruleus output through the sympathetic dilator pathway. The recovery time constant τreturn is therefore a non-invasive dynamical index of LC arousal state, inaccessible to instruments that treat the pupil as a static aperture. This paper develops the computational framework that makes that index rigorous. The pupillary light reflex is modelled as a nonlinear delay-differential equation following Longtin and Milton [5], with a gain structure that pins the linearised loop gain exactly to a single dimensionless parameter at every operating point. The exact analytical solution of the characteristic equation replaces the classical zero-delay approximation, relocating the bifurcation threshold and providing a calibration curve whose sensitivity is concentrated precisely where arousal resolution matters most. Simulation confirms that the recovery time constant tracks this curve closely under realistic noise conditions, and a spectral analysis of continuous pupil traces cross-validates the bifurcation geometry through an independent computational pathway. A three-variable non-invasive instrument architecture is specified, combining resting diameter, recovery dynamics, and chromatic post-illumination response into a self-consistent output whose internal validity is guaranteed by the physiology rather than imposed by design. All results are computational; three falsifiable predictions are identified for empirical validation.

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