Emmetropia approach dynamics with diurnal dual-phase cycling

Numerical experiments are performed on a first order exponential response function subjected to a diurnal square wave visual environment with variable duty cycle. The model is directly applicable to exponential drift of focal status. A two-state square wave is employed as the forcing function with high B for time H and low A for time L. Duty cycles of (1/3), (1/2) and (2/3) are calculated in detail. Results show the following standard linear system response: (1) Unless the system runs into the stops, the ready state equilibrium settling level is always between A and B. The level is linearly proportional to a time-weighted average of the high and low states. (2) The effective time constant t(eff) varies hyperbolically with duty cycle. For DC = (1/3) and t1 = 100 days, the effective time constant is lengthened to 300 days. An asymptote is encountered under certain circumstances where t(eff) approaches infinity. (3) Effective time constants and steady state equilibria are independent of square wave frequency f, animal time constant t1, magnitude and sign of A & B, and diurnal sequencing of the highs and lows. By presenting results on dimensionless coordinates, we can predict the drift rates of some animal experiments. Agreement between theory and experiments has a correlation coefficient r = 0.97 for 12 Macaca nemestrina eyes.