Stability and bifurcation analysis of a pupillary light reflex model

In this paper, we investigate a non-linear, time delayed model of Pupillary Light Reflex (PLR). We take into account its key performance metrics like stability, convergence and robustness. Using time and frequency domain analysis, we study its stability properties and offer guidelines on parameter values that guarantee local stability. Trade-offs between system parameters are explored with the help of stability charts. The values of parameters for which oscillatory and non-oscillatory convergence occur are analysed. We prove that each parameter can induce a loss of stability via a local Hopf bifurcation. Further, the stability of the ensuing limit cycles are characterised analytically using normal forms and the centre manifold theorem. Bifurcation diagrams accompany the analytical results. We establish that the limit cycles generated are always unstable. It reveals that the pupillary reflex model becomes difficult to control once it loses its local stability. The robustness of the model is measured for uncertainities in parameter values. Our work provides design-friendly guidelines to ensure stability and achieve desired level of performance and robustness.