Fig 1.
Phase estimates for users of Entrain.
Motion data are processed through the mathematical model of the human circadian clock developed by Jewett et al. [5]. The sample includes 122 subjects with at least five consecutive data days and seven days with no travel. (A) Average phase estimates (in UTC) in different GMT Offsets. Each dot represents the average phase estimate for each GMT Offset. The dashed line shows the best-fit line to the data points across all GMT offsets. The best-fit line suggests the difference between phase estimates is consistent with the time zone difference. (B) Box-plot of phase estimates (in UTC) across continents. (C) Box-plot of phase estimates (in local time) between genders. (D) Box-plot of phase estimates (in local time) across different age categories.
Fig 2.
Mood assessments for users of Entrain.
The mood assessment consists of 20 items, with ten items measuring positive affect and ten items measuring negative affect. Each item is rated on a five-point scale (1 indicating very slightly, five indicating extremely). The y-axis, ‘Positiveness,’ represents the difference between the ten items of positive affect and ten negative affect items. (A, B) A summary of mood assessments of 680 users completed within a week after trips shows results regarding gender (A) and age (B). (C) This figure shows mood assessments with predicted circadian phases. Seventy-one subjects submitted at least five consecutive days of motion data before filling out the mood assessment. A quadratic trend is observed in mood assessments with different phase estimates. (D) Thirty-one subjects completed mood assessments within one day after trips to assess mood for different traveling directions. This figure shows that jet lag, as measured by mood, is worse when flying east. (E, F) The relationship between subjects’ moods and the extent to which they followed their optimal schedule is shown. Twenty-eight subjects submitted their motion data during the recommended schedule for phase adjustments. For each episode of the recommended light schedule (i.e., avoiding/receiving light), the subject is considered slightly active if he/she is active under 30% of the time during the scheduled period, and highly active otherwise. This figure suggests that users who follow the recommended light schedules feel less jet lag.
Fig 3.
Optimal schedule deployed in real life.
(A) Activity data (black line) from one subject who took three trips (shaded in blue) in 25 days. Optimal schedules (grey for avoiding light, and yellow for receiving light) to adjust to the new time zones were computed for each trip. We can see the subject generally followed the recommended optimal schedule for his/her first and last trip, but not for the second trip. (B) A sample of 147 Entrain users who submitted their motion data during trips shows how the recommended schedules were followed each hour of the day. Each yellow/gray bar represents the average percentage of active time during the scheduled light/dark period at each hour of the day. Unsurprisingly, regardless of the recommended light-receiving/light-avoiding schedule, the activity pattern peaks during the day and drops during the night. (C) This figure shows the extent to which the subjects followed the recommended schedule for different time durations. Each yellow/gray bar represents the average percentage of active time at each recommended duration of the light/dark period. As the recommended period of darkness increases, so does the percentage of active time under recommended darkness. (D) This figure shows how the recommended schedules were followed as subjects traveled across different time zones, where positive time zones represent traveling east. Each yellow/gray bar represents the average percentage of active time during the recommended light/dark period for the total number of time zones crossed. (E) This figure shows the percentage of active time under recommended darkness under the recommended light for the total number of time zones crossed. As can be seen, the optimal schedule is harder to follow when flying west on short trips. The low compliance with the schedule may also occur because people do not feel the need to observe light and dark recommendations for short phase delays.
Fig 4.
Optimal schedules for entrainment to 12-hr shifts are simulated with different types of noise for 30 hypothetical subjects. (A-C) Phase trajectories with other sources of error were individually studied. The human circadian pacemaker acts as a limit cycle oscillator formed by two variables, x, and xc. The variable x represents the core body temperature, a phase marker of the human circadian rhythm, and the variable xc is required to achieve a limit cycle mathematically. The noiseless, optimal trajectory is plotted in the dark dashed line, while noisy trajectories are plotted in a lighter gray. The start and end of the trajectories are marked in green and red, respectively. Three sources of error were considered: initial conditions (i.e., starting circadian state), light levels in lux, and switch time (the times at which light is either switched on or off). For each source of error, the noise was sampled randomly from a normal distribution with mean 0 and standard deviation σ, where the standard deviation in each case is σinitial condition = (σamplitude, σphase) = (0.1,1), σlight = 1000 lux, and σswitch time = 2 hours respectively. (D-F) 24-hour snapshots of the circadian phase with all different types of noise acting together. Graphs from left to right represent the increasing magnitude of noise added to switch times (σswitch time = 2, 3, 4 hours), with a fixed magnitude of noise added to initial conditions and light levels σinitial condition = (σamplitude, σphase) = (0.1,1), σlight = 1000 lux). Predicted core body temperature minima (CBTmin) are plotted against the schedule of optimal light exposure, where the color yellow and black represent bright light exposure (10000 lux) and darkness (0 lux), respectively. Predicted CBTmin under the optimal schedule without noise is marked in red circles, while predicted CBTmin of 30 hypothetical subjects under schedule with noise is plotted in blue circles.
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.
Fig 6.
Comparison of optimal and accessible lighting schedules for different light levels.
(A) (Left) Optimal light schedules are computed with no constraints on light/dark duration, with a maximum lux value of 3000 lux. (Right) Optimal light schedules are calculated with constraints, where we require the times spent in light to be greater than five hours in duration and the times in dark to be less than ten hours. Each vertical slice of light and darkness is a schedule, with the colors defaulting to the background gray when a person following the schedule is sufficiently close to entrained. The time to entrainment is nearly identical despite markedly different durations of darkness for some shifts. (B) (Left and Right) Unconstrained and constrained optimal schedules for a maximum lux value of 500 lux.
Fig 7.
Effect of stopping an optimal schedule early.
(A) Mean phase difference between the final point of the prematurely ended the optimal schedule and the target over the next 24 hours. The function is nonlinear, with a sharp change in the mean phase difference (and time to entrainment) occurring at approximately the halfway point in the schedule. The final difference is non-zero due to the tolerances set for convergence in the Switch Time Optimization algorithm [25]. (B) Visualizing the trajectories by sampling every 24 hours in phase space. The green circle marks the starting point. Lighter trajectories with brighter red circles as the final states correspond to more of the optimal schedule completed. A triangle marks each day’s predicted phase.