Optimal adjustment of the human circadian clock in the real world
Fig 5
Simulated trajectories from real data, optimal schedule, and nearest optimal schedule.
To illustrate how optimal schedules and closest optimal schedules work, all four figures here select a target zone (dashed red circle) in the upper left quadrant. The target zone represents the point at which a trajectory is within a tolerance of the perfectly entrained target point. The variables, x, and xc, form a limit cycle, representing the human circadian pacemaker with 24.2 hours. The predicted circadian phase of each day is marked with a triangle (Δ). Green dots mark the start of trajectories, and the end of trajectories after sufficient entrainment are marked in red. (A) Sample trajectories are simulated by the self-reported lighting history of Entrain users with the circadian model. As can be seen, by the number of triangles and the trajectories’ winding direction, phase shifting in the real world is inefficient. (B) Sample trajectories of optimal schedules starting from 24 initial states that are evenly sampled on the limit cycle are plotted in the (x, xc) space. We can observe that the optimal trajectories are almost straight lines, indicating that the entrainment is efficient. (C) Trajectories follow the nearest optimal schedule from randomly selected starting points. A subset of twelve starting points is randomly chosen from the starting points on the limit cycle in (B) added with noise. The noise is randomly sampled from a normal distribution with mean 0.5 and standard deviation 0.25. Following the nearest optimal schedule introduces error, which results in some final states not arriving in the target zone. (D) Trajectories follow the nearest optimal schedule from randomly selected starting points with schedule updates occurring at regular intervals by repeating the method of the nearest optimal schedule. As can be seen, updating the schedule at regular intervals corrects the error and results in all final states landing in the target zone.