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The authors have declared that no competing interests exist.

Conceived and designed the experiments: KS DBF. Performed the experiments: KS DBF. Analyzed the data: KS DBF. Contributed reagents/materials/analysis tools: KS DBF. Wrote the paper: KS DBF.

Jet lag arises from a misalignment of circadian biological timing with the timing of human activity, and is caused by rapid transmeridian travel. Jet lag's symptoms, such as depressed cognitive alertness, also arise from work and social schedules misaligned with the timing of the circadian clock. Using experimentally validated mathematical models, we develop a new methodology to find mathematically optimal schedules of light exposure and avoidance for rapidly re-entraining the human circadian system. In simulations, our schedules are found to significantly outperform other recently proposed schedules. Moreover, our schedules appear to be significantly more robust to both noise in light and to inter-individual variations in endogenous circadian period than other proposed schedules. By comparing the optimal schedules for thousands of different situations, and by using general mathematical arguments, we are also able to translate our findings into general principles of optimal circadian re-entrainment. These principles include: 1) a class of schedules where circadian amplitude is only slightly perturbed, optimal for dim light and for small shifts 2) another class of schedules where shifting occurs along the shortest path in phase-space, optimal for bright light and for large shifts 3) the determination that short light pulses are less effective than sustained light if the goal is to re-entrain quickly, and 4) the determination that length of daytime should be significantly shorter when delaying the clock than when advancing it.

When our body's internal timekeeping system becomes misaligned with the time of day in the outside world, many negative effects can be felt, including decreased performance, improper sleep, and jet lag. When misalignment is prolonged, it can also lead to serious medical conditions, including cancer, cardiovascular disease, and possibly even late-onset diabetes. Rapid readjustment of our internal daily (circadian) clock by properly timed exposure to light, which is the strongest signal to our internal circadian clock, is therefore important to the large proportion of the population which suffers from misalignment, including transmeridian travelers, shift workers, and individuals with circadian disorders. Here we develop a methodology to determine schedules of light exposure which may shift the human circadian clock in the minimum time. By calculating thousands of schedules, we show how the human circadian pacemaker is predicted to be capable of shifting much more rapidly than previously thought, simply by adjusting the timing of the beginning and end of each day. Schedules are summarized into general principles of optimal shifting, which can be applied without knowledge of the schedules themselves.

Modern society requires individuals to be awake and alert at times that conflict with their internal circadian (∼24-hour) timekeeping systems. In year 2012 over 60 million Americans traveled overseas, subjecting themselves to long periods of circadian mistiming, impaired sleep, and low performance

Light is the strongest signal to the human circadian system

Accurate mathematical models for the effect of light on the human circadian system are available

Mathematical models can, in theory, be analyzed to determine optimal schedules, or schedules which outperform all others

Here we describe a mathematically robust method, which uses existing mathematical models

Using this approach, we determined over 1,000 schedules that optimally re-entrain, without the limits on the length of schedules imposed by prior studies. Moreover, while previous work often assumed that the light available for shifting was 10,000 lux or higher, we considered many light levels, including those found indoors.

Each optimal schedule for

In

Predicted circadian phase, indicated by simulated core body temperature minima (CBTmin, magenta triangles), is plotted against the pattern of exposure to bright light (10,000 lux, yellow), moderate light (100 lux, white), dim light (5 lux, gray), and darkness (0 lux, black). Predicted CBTmin under noisy light levels (See supplemental

To study the effects of these schedules, we also plot the process of re-entrainment in terms of both phase and amplitude. Thus, we produce experimentally measurable predictions for phase-amplitude resetting maps (PARMs)

Circadian phase is plotted in degrees, with 0° corresponding to entrained CBTmin (before the schedule shift occurs), while amplitude is measured on the radius. A 6 hour advance would be indicated by a shift to 90°, a 12 hour shift to 180°, and a 6 hour delay to 270°. The sleep/dark region is indicated by the shaded regions on the PARMs. The timing of entrained CBTmin in the new time zone is indicated by the dotted line. Figures (

Our proposed schedule for optimal (minimum-time) complete re-entrainment to a 12-hour time zone shift takes approximately 4 days, whereas the five previously proposed schedules require more than 7 (see

Schedule | Time to complete re-entrainment (days) | Time to partial re-entrainment (days) |

Slam shift | >13 | 13 |

Sack et al. |
9 | 4 |

Waterhouse et al. |
13 | 3 |

Eastman et al. |
>13 | 6 |

Dean et al. |
7 | 3 |

Optimal schedule (complete) | 4 | 3 |

Optimal schedule (partial) | 13 | 2 |

Since real-world light levels are highly variable, we added variability in the lighting conditions to mimic the natural environment (See supplemental

We next computed optimal schedules for complete re-entrainment in minimum time to an 8 hour advance and an 8 hour delay (See

Predicted core body temperature minima (CBTmin, magenta triangles) are plotted against the pattern of optimal exposure to bright light (200 lux–10,000 lux, yellow), moderate light (100 lux, white), and darkness (0 lux, black). Predicted CBTmin under noisy light levels (See supplemental

Reducing the maximum available light level ten-fold to 1,000 lux yielded optimal schedules similar to the schedules for 10,000 lux, except that complete re-entrainment required an additional day. Reducing the maximum light level an additional two-fold to 500 lux added another day to each schedule. While the daily light exposure in schedules for the 12 hour shift and 8 hour advance changed little with reduced light, the 8 hour delay showed very little amplitude suppression (3G) compared to when more light was available (3A and 3D). Reducing the maximal light level to 200 lux required an additional two to three days to shift. At this lower light level, no amplitude suppression occurred for the 8 hour advance or the 8 hour delay. When the maximum available light was further reduced to 100 lux, no amplitude suppression occurred for any shift. These transitions are clearly visible in the corresponding PARMs (See supplemental

We next found optimal schedules for complete re-entrainment to all phase shifts with maximum light levels of 10,000, 1,000, 500, 200 and 100 lux consisting of approximately 1,000 optimized schedules. These are summarized in

This plot shows the pattern of bright light (200 lux–10,000 lux, yellow), background light (100 lux, white), and darkness (0 lux, black) under which the clock is optimally reset from the corresponding initial phase. If a vertical line is drawn on the plot, then the pattern of light and dark along this vertical line is the optimal schedule for resetting the clock from the corresponding initial phase. Figure (

The format used in

The meaning of “initial phase” measured on the horizontal axis comes from the following interpretation. Without loss of generality, we fix our destination time zone, and consider shifts from all possible time zones. Each vertical slice corresponds to an optimal schedule for complete re-entrainment from the initial phase (where the slice intersects the horizontal axis) to phase zero. Days −2 and −1 show a 16∶8 light-dark (LD) cycle of 100 lux (dim home or office lighting) in the original time zone. At the beginning of day 0, the transition to the new time-zone begins. An optimal schedule for re-entrainment is presented, then once it is finished, the LD cycle in the target time zone takes over. Again, here black corresponds to darkness, white to dim light (100 lux), and yellow to bright light (200–10,000 lux depending on the intensity). We also simulate the predicted circadian response to these schedules, displaying both circadian phase and amplitude in

The format is exactly the same as

From this plot (

A second class of schedules is seen when the maximal light levels are lower (e.g. 200 or 100 lux) or for smaller (<8 hour) phase shifts with 500 lux or 1000 lux. These optimal schedules often shift with minimal changes to the preferred circadian amplitude. We call this type of shifting limit-cycle shifting (LCS). In LCS, light stimuli are presented which maximally advance or delay the circadian clock while near (but not on) the limit cycle; the next day's light stimulus is the same as the previous day's except that is presented at the appropriate time considering the phase shift predicted to occur. This is supported by

We simulated the PRCs to all possible one-pulse stimuli for a variety of different light levels. For each light level, two stimuli were selected: the one producing the greatest advance and the one producing the greatest delay. The model was kept in total darkness before the stimulus was administered. Resulting phase shifts were measured using the concept of isochrons

An important feature of LCS schedules is that the daily stimulus to delay the clock is much shorter than the stimulus to advance the clock. Moreover, there appears to be region of phases where no light appears in either stimulus. This goes against the predictions given by

Repeating these computations on the model described in

Finally, while we present the schedules in a form (i.e.

In this study, we have found locally optimal schedules which completely re-entrain the human circadian pacemaker in minimum time. The methodology we propose can determine the optimal schedules directly from the model, without any additional assumptions, and can create schedules which outperform any which are not locally optimal. Schedules are efficient, easy to follow, and robust to changing light levels and inter-individual differences. Our schedules are based on not one but two widely used mathematical models:

We find that MPS schedules are better when the phase shift is large and bright light is available, and LCS schedules are better when the phase shift is small and only dim light is available. The reasons are as follows. In general, schedules attempt to take the pacemaker on the shortest path to re-entrainment. For this light may be used to decrease circadian amplitude, and in doing so, is opposed by the effects of the pacemaker attempting to return amplitude to its original level (limit cycle). This is analogous to the problem of pushing a ball over a hill. The circle around the hill is like the phases of the clock along the limit cycle, and the steepness of the hill is the effect of amplitude recovery, which pushes the oscillator to the limit cycle. When light levels are too low, the amplitude recovery is stronger than the effect of light, and light cannot move the clock on most direct path between two points on the limit cycle.

Our results challenge previously held assumptions about efficient phase shifting of the human circadian clock. It has been previously suggested by many authors that a schedule that passes close to the phase singularity

We also find that the dynamics of circadian photoreception in humans has a large impact on phase shifting in minimum time. While short pulses of light can give nearly as much signal to the circadian pacemaker as continuous light

Finally, we describe how schedules for complete re-entrainment in minimum time can be used to create more practical schedules for the treatment of jet lag. This is done using optimal schedules to rapidly shift CBTmin into the sleep/dark region, specifically by shifting it to the beginning of the region. This may facilitate better sleep quality in the new time zone, and may resolve many of the symptoms associated with jet lag more quickly than schedules for complete re-entrainment.

Circadian misalignment due to jetlag is a major problem for modern society. The optimal schedules presented here, perhaps especially the schedules for partial re-entrainment, bring us closer to designing schedules which may help travelers re-entrain quickly. More importantly perhaps, the principles described in this manuscript could be used to compute and design customized schedules which help individuals re-entrain while minimizing jet lag and performance lapses in practical settings, such as shift work, where many parameters such as the amount of exposure to bright light or the amount of darkness/sleep are constrained. Moreover, the method could be generalized in a straightforward way to multiple control inputs in addition to light, such as the timing of sleep, exercise, or pharmacological treatments, further accelerating re-entrainment. It could also be applied to multi-level oscillator models such as

We were pleased to find that the schedules we present are simple to follow, in the sense that they involve only a single daily light exposure, and that they are predicted to yield uniform results even in the presence of unpredictable factors. We found a significant effect of the circadian phototransduction system on schedules, and that some schedules match aspects of previous recommendations, e.g. avoiding morning light

Our methodology to compute optimal schedules consists of two major contributions. First, we define the re-entrainment problem in terms of optimal control theory. This includes computing the

The models we use

The constraint is defined as follows. Unlike previous works, we explicitly compute the isochrons of the model in the form of a function

The top plot shows the entraining stimulus (LD-cycle) and the leftmost plot, the entrained limit cycle corresponding to this stimulus. Here pink corresponds to day and black to night. The middle plot shows the oscillations in the first coordinate of the entrained limit cycle. The last plot shows the phase of periodic stimulus or, equivalently, of the entrained oscillator. All significant features are labeled with the appropriate notation (See

The top plot shows the entraining stimulus (LD-cycle) to the oscillator, plotted on a log scale, shifted by

The cost is defined in the following way. Since we would like to minimize time

Once we have an optimal control problem to solve, we compute its solution using a numerical algorithm. We use a novel modification of a numerical method called the Switch Time Optimization method

When this algorithm converges – in the sense that the guess can no longer be improved – we find that the solution satisfies a set of local optimality conditions called Pontryagin's Minimum Principle (see supplemental

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We thank Willard Larkin for a careful reading of this manuscript.