7

##### RESULTS AND DISCUSSION

Figure 2(a) shows the strong effect duty cycle has on the effective time constant of the system. Figure 2(b) shows that the equilibrium settling level is simply a time-weighted average of the highs and lows of the applied square wave environment. Figure 3, with animal data from Young (1961) arid a time constant t\ of 100 days, correlates at the level r = 0.97 for 12 age matched Macaca nemestrina eyes from six adolescent monkeys. Theory agrees with experiment plus or minus 0.06 diopters, i.e. with greater accuracy than one can currently measure experimentally. Thus, we say with confidence the drift dynamics are exponential.

########## Applications

As a practical example of these equations, say you are observing a 4 diopter refractive state R(t) subjected to a 3D and 4D square wave visual environment with duty cycle of (1/3). With a normal time constant of 100 days, this system will take approx. 300 days to reach the (1/e) point, hence the origin of the effective time constant. With a re-scaling of the system boundary conditions, based on the time-weighted average of the applied environment, the time constant re-scales to 100 days. Independent of duty cycle, an exponential response always results. Time constant is always preserved. But, as a practical matter, the system takes longer to influence with low impulse cycles.

In terms of practical real-time applications, notice that 4-6 h per day one way or the other, i.e. a fraction of one duty cycle, can make the difference between 1 year and 3 years in terms of achieving an objective level. Ku and Greene (1981) present similar time-weighted average results appropriate during the initial dynamic stage (i.e. t < to) for variable duty cycle loading of rabbit eyes; in this case, theory and experiment agree with correlation r = 0.91.

Focusing during sleep is a matter of experimental interest. Events during the night are not known. We can assume, based on preliminary experiments (see Introduction), that the eye resorts to a compensatory offset, or "dark focus" value of 0.5 D of hyperopic reserve. The net optical impulse of the sleep cycle should approximately balance the environmental impulse accumulated during the waking hours to maintain steady state level. If this is not the case, theory predicts that R(t) will drift to a new level. In the absence of experimental data while the subject is unconscious, we can invent a Conservation of Impulse law to indirectly subsume the night time activities. Results presented here are sufficiently general to accept any level .

Note added after proof. added after proof. Another practical application of this mathematical model is predicting the effect of bifocal or progressive addition lenses ( PALs ) on the progression rate of myopia. In this case, the difference between the near field ( A diopt.) and far field ( B diopt. ) values of the environment are reduced in magnitude by approximately 2 diopters, the value of the ( + ) reading Add in the lower segment of the frame. Theory predicts this should slow the rate of myopia progression [12] [19].

########## General results

Effective time constants and steady state equilibrium levels are independent of square wave frequency /, animal time constant fl, magnitude and sign of the high and low states A and B, and even random diurnal sequencing of the highs and lows. Presenting results on dimensionless coordinates allows prediction of different animals with different time constants in variable environments. In general, the observed and calculated diopter drift dynamics are consistent with standard control theory and first order linear system response.

7