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Circadian System Modeling and Phase Control

Jiaxiang Zhang, Andrew Bierman, John T. Wen, Agung Julius, Mariana Figueiro

Rensselaer Polytechnic Institute

Troy, NY 12180

{zhangj16,bierma2,wenj,juliua2,figuem}@rpi.edu

Abstract—Circadian rhythms are biological processes found

in all living organisms, from plants to insects to mammals that

repeat with a period close to, but not exactly, 24 hours. In

the absence of environmental cues, circadian rhythms oscillate

with a period slightly longer or shorter than 24 hours. The 24-

hour patterns of light and dark are the strongest synchronizer

of circadian rhythms to the solar day. Circadian disruption

resulting from lack of synchrony between the solar day and

the internal master clock that regulates and generates circadian

rhythms had been linked to a variety of maladies. Circadian

disruption, as experienced by night shift workers or by those

traveling multiple time zones can lead to lower productivity,

digestive problems and decreased sleep efficiency. Long-term

circadian disruption has been linked to serious health problems,

such as increased risk of cancer, cardiovascular disease, diabetes

and obesity. Biochemical and empirical mathematical models

describing the circadian clock and its response to light input

have been developed by various research groups. Biochemical

models describe the kinetics of the interaction between different

proteins and may be of high order depending on the complexity

of the model. Empirical models are based on nonlinear oscilla-

tors, such as the van der Pol oscillator, and are, therefore, much

simpler. Though empirical models do not have a biochemical

basis, it has been shown that they do represent the averaged

asymptotic behavior of the biochemical models. In this paper,

we analyze a simple empirical model proposed by Kronauer

and colleagues and discuss how light control may be used

to promote circadian entrainment. In contrast to most of the

existing approaches, which are based on phase response curves,

we propose a feedback-based system. Through simulation, we

show that the recovery of a 12-hour jet lag can be shortened

from 7 days to 2.5 days.

I. INTRODUCTION

The Earth rotates on its axis with a regular and predictable

24-hour pattern of daylight and darkness. Terrestrial species

have adapted to this daily pattern by evolving biological

rhythms that repeat at approximately 24-hour intervals. These

are called circadian rhythms (circa=approximately; die=day).

For humans, circadian rhythms are regulated and generated

by a master clock located in the suprachiasmatic nuclei

(SCN) in the hypothalamus in the brain. The SCN governs

a wide range of biological cycles, from cell division, to hor-

mone production, to behavior (e.g., sleep-wake) that, when

synchronized with the natural light/dark cycle, enables the

organism to entrain these cycles to its particular photic niche

(diurnal or nocturnal) and to its location on Earth. In humans,

examples of circadian rhythms include the sleep/wake cycle,

hormone production and levels of daytime/nighttime perfor-

mance and alertness. Lack of synchrony between the master

clock in the brain and the external environment, referred to as

circadian misalignment, can lead to circadian disruption, with

the attendant detriments ranging from increased sleepiness

during the day, lower productivity, gastrointestinal disorders,

to long-term health problems such as increased risk for

cancer, diabetes, obesity, and cardiovascular disorders [1],

[2]. Rotating-shift work, transcontinental flights, irregular

sleep patterns, all of which will lead to irregular light/dark

exposures can all contribute to circadian disruption.

The SCN exhibit sustained oscillations in vivo and in

vitro. Numerous mathematical models have been proposed

to describe the oscillations of the master clock. For example,

based on studies in Drosophila Melanogaster (commonly

known as fruit fly), a 10-state model of the kinetics of the

circadian clock gene network has been proposed [3] and has

been used for circadian system control [4], [5]. It is important

to note, however, that although applicable across species,

clock genes in mammals are not exactly the same as in fruit

flies.

An alternative approach is to use a nonlinear oscillator,

such as the van der Pol oscillator, as an empirical model for

the circadian rhythm. Such approach has long been proposed

[6], and more recently it was used by Richard Kronauer

et al. [7]. The Kronauer model describes the relationship

between light stimulus and the oscillation of human core

body temperature, which is an acceptable phase marker of

the circadian system. In the Kronauer model, the circadian

system is represented by a modified Van der Pol oscillator

with a 24-hour period. Light stimulus is first converted into a

driveˆB through a light stimulus dynamic processor, process

L. This is analogous to the bleaching of photopigments in

retinal photoreceptor by photons, which remain unusable for

further photon response until they are regenerated. The drive

ˆB feeds into the process P which represents the circadian

pacemaker. The driveˆB is first modulated to B which is the

input of the modified van der Pol oscillator. The oscillator

output is related to the oscillation of core body temperature

which is used as a phase marker of the circadian system.

As argued in [8], the empirical model may be considered

as the asymptotic case of the biochemical model in an

averaged sense. A reduced order modeling approach for the

biochemical model has also been proposed in [9].

The effect of light stimulus is commonly characterized

by the phase response curve (PRC), which plots the steady

state phase shift as a function of the time of the day at

which a light pulse with a given amplitude and duration is

applied. The PRC may be generated experimentally using test

subjects or numerically with a chosen simulation model. An

open-loop circadian entrainment method has been proposed

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based on the PRC constructed from the Kronauer model [10]

to design a light-dark pattern for jet lag treatment. However,

open-loop control does not address the difference between

individuals and the uncertain ambient light disturbance. In

[5], the 10-state circadian oscillation model of Drosophila

is used to construct the PRC. A closed-loop phase shifting

control based on the model predictive control method is

developed based on this PRC. With only the timing of the

light pulse adjustable (intensity and duration are constant),

the simulation result shows that a 12-hour phase shift is

achieved within 3.5 days.

In this paper, we focus on the Kronauer model. We first

present an analysis of its behavior and properties, and then

propose a feedback control strategy for the light input to reg-

ulate the oscillation. By taking advantage of the higher phase

shifting efficiency at lower circadian oscillation amplitude,

we further modify the controller to a two-step design for

faster circadian entrainment. Though the stability analysis is

only heuristic at the present, we show through simulation that

our proposed method can shorten 12-hour jet lag recovery

from 7 days to 2.5 days with 10,000lux (equivalent to full

day light but not direct sun) controlled light input.

II. MODELING OF CIRCADIAN RHYTHM

A variety of models have been proposed. Based on the

study of Drosophila, a model based on Per and Tim protein

dynamics was proposed in [3]. Under some simplifying

assumptions, a 10-state affine model is obtained. This model

can exhibit a rich set of behaviors from limit cycles to chaotic

orbits. The model has been used in some lighting control

research to regulate the phase of the oscillation [4], [5]. It is

recognized that from the phase control perspective, the model

may be reduced. In [9], a 3-state reduced order model was

obtained based on proper orthogonal decomposition. More

complex model involving the interaction of multiple proteins

and gene expressions have been proposed [11], [12], but such

complex model has not been used in control analysis.

An alternate approach was proposed by Kronauer for an

empirical model relating light input to core body temperature

output [7]. This model couples a first order lighting stage,

L-process, followed by a second order van der Pol type of

oscillator, called the P-process. The overall model is given

by

˙ n

=

=

60[α(1−n)−βn], α = α0(I/I0)p

f0(z)+ f1(z)u

(1)

˙ z

(2)

where

z =?

xxc

?T, u = Gα(1−n), f0(z) = Az+Bg(BTz)

µ/31

−(24/(.99729τx))2

g(x) =π

f1(z) =π

12

1

qxc+kx

and I is the intensity of the light input. The numerical

parameter values given in [7], [13] are:

A =π

12

?

0

?

, BT=?

10

?

12µ?(4/3)x3−(256/105)x7?

?T(1−0.4x)(1−0.4xc)

?

µ = 0.13, τx= 24.2, q = 0.33, k = 0.55

α0= 0.05, β = 0.0075, I0= 9500, G = 33.75, p = 0.5.

Note that this model represents the average circadian rhythm

behavior. For a particular organism, τx, which determines the

period, would deviate from this averaged value.

The Kronauer model is attractive as the oscillator portion

is second order and lends itself to more intuitive phase plane

analysis. The model of course does not exhibit the same rich

range of behaviors as the 10-state model, for example, chaos

cannot occur. However, it does capture the essential feature

of the circadian rhythm, and we shall use it as a baseline for

the development of the phase control algorithms.

III. ANALYSIS OF KRONAUER MODEL

For a fixed intensity light input I, n→α/(α+β) with rate

60(α +β). In this paper, we shall assume that the L process

is sufficiently fast (i.e., β is sufficiently large) so that we can

just focus on the dynamics of the P process. As n converges

to a steady state, the corresponding u → Gαβ/(α + β),

which is a monotonically increasing function in α. There-

fore, we will consider u as the effective non-negative input

variable with maximum value Gβ (corresponding to infinite

light intensity).

When there is no light input, i.e., u = 0, there is only one

equilibrium at the origin, z = 0. The linearized dynamics is

governed by A which has two unstable complex eigenvalues.

This means that the origin is an unstable center, and trajecto-

ries near the origin will spiral outward. When x is large, g′(x)

becomes negative. It may be shown that vector field points

inward. By Poincar´ e-Bendixson Theorem, there exists a limit

cycle around the origin. We can estimate the period and

amplitude of the limit cycle by using the describing function

method (also known as harmonic balance) [14], [15]. The

system may be written as a feedback interconnection of

G(s) = BT(sI −A)−1B and an odd nonlinearity g(·). The

Bode plot of G(s) shows a peak at ω = 0.26rad/hr which

corresponds to a period of 24 hours. The sharp drop from

the peak means that the effect of higher harmonics will be

attenuated, thus, the describing function method has a good

chance of making a reasonable prediction of the limit cycle.

The describing function approach approximates the periodic

solution x(t) by asinωt. Since g is a static function, g(x(t))

is also periodic. Retaining only the fundamental harmonic of

x(t) and g(x(t)), we may replace g by its describing function

[14], [15]:

η(a) =π

12µ

?

a2−4

3a6

?

.

(3)

We then solve for a such that A + Bη(a)BThas purely

imaginary eigenvalues, jω. For the Kronauer model, a = 1,

and T = 2π/ω = 24.13hr.

IV. CIRCADIAN RHYTHM CONTROL

The natural light-dark pattern involves the switching of I:

a constant value of I during the light state and I = 0 in the

dark state. The corresponding steady state u would switch

between 0 and umax= Gαβ/(α +β) where α is given by

(1). Figure 1 shows the result of a 12-hour light-dark cycle

which maintains the 24-hr period while introducing a phase

shift of about 1 hour as compared with the free running

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oscillation. This synchronization of the oscillation with the

external light-dark pattern is called entrainment.

050 100150 200

−1

−0.5

0

0.5

1

time (hr)

x

Period = 24 hoursfree running

12 hour day light

light input, u

−1.5 −1−0.50

x

0.51 1.5

−1.5

−1

−0.5

0

0.5

1

1.5

xc

Period = 24 hours

free running

12 hour day light

Fig. 1. Effect of 12-hour light-dark pattern

The light input may be manipulated, through artificial

lighting, to achieve different objective, such as:

• Entrainment Control: In the presence of disturbances,

e.g., spurious light, sleep disorder, or other stimuli as

in the case of shift workers, that affect the circadian

rhythm, light control may be used to ensure regular

period and phase of the circadian oscillation.

• Phase Control: To achieve specified phase shift, which

is useful to address jet lag recovery, light control may be

useful to reduce the recovery (re-synchronization) time.

A. PRC based Phase Control

The phase response curve, or PRC, obtained experimen-

tally or numerically, may be used to develop entrainment

strategy. However, the PRC is highly dependent on the pulse

amplitude and duration. In the case of a pulse train input

or even a single pulse with sufficient strength (product of

amplitude and duration), the numerically generated PRC is

a slanted straight line (also as shown in [16]). At lower light

pulse strength, PRC can take on different shapes.

If the light pulse strength is large enough, one may even

be able to change the period of the circadian rhythm (e.g., for

submariners) [17]. Figure 2 shows the effect of light pulses

at 30hr intervals. When there is insufficient light input (6-

hr light and 24-hr darkness), there is no entrainment. With

sufficient light input (12-hr light and 18-hr darkness), the

oscillation is entrained at 30-hr period.

0100 200300 400500 600

−1.5

−1

−0.5

0

0.5

1

1.5

time (hr)

x

Period = 30 hours

free running

3 hour day light

light input, u

0100200 300400500600

−1.5

−1

−0.5

0

0.5

1

1.5

time (hr)

x

Period = 30 hours

free running

12 hour day light

light input, u

Fig. 2.

light pulse (left), circadian rhythm entrainment is not achieved. At 12-hour

light pulse (right), circadian rhythm is entrained.

Entrainment control using 30-hour light/dark pattern. At 3-hour

To change the phase, we will focus on using light in-

puts to change the phase of the free running oscillation.

The approach is equally applicable to phase change in the

presence of a given light-dark pattern. The PRC approach has

been proposed in [4], [5], except the PRC is generated with

a single light pulse with specified amplitude and duration

applied at different time. The resulting steady state phase

shift, φ, is then plotted as a function of the time of the

applied light pulse, T: φ = h(T). If the desired phase shift

is within the range of h, then T may be directly selected.

Otherwise, a simple iterative approach may be applied to se-

quentially reduce the phase error to zero. A multi-parameter

PRC may also be generated by varying the magnitude and

duration of the light pulse. The resulting phase shift is then

φ = h1(T,A,Td), and a constrained minimization may be

performed to select (T,A,Td) to sequentially and optimally

reduce the phase error to zero. This type of approach has

been popular and used in various forms by many researchers

[3].

B. Phase Plane Analysis

We now look more closely how light input would affect

the phase by considering the effect of a light pulse on the

phase of the free running oscillation through a phase plane

analysis. As shown in Figure 3, the pulse input kicks the state

off the limit cycle to za, resulting in an immediate phase shift

of ∆φa. The resulting orbit then converges back to the limit

cycle, with an additional phase shift of ∆φb.

Fig. 3.Effect of light pulse on phase

Transient Process: The phase shift, ∆φa, due to a light pulse

u of magnitude J and duration ∆T, may be estimated from:

cos(∆φa) =r2

a+r2

2rarb

b−r2

c

(4)

where

ra

=

=

?z+(f0(z)+ f1(z)J)∆T?

?z+ f0(z)∆T?

?f1(z)?J∆T.

rb

rc

=

The corresponding time shift is ∆Ta= (24/2π)∆φa.

Recovery Process: The phase shift due to the convergence

back to the limit cycle may be computed numerically by

running simulations using initial conditions all round the

limit cycle. The resulting phase shift map as a function of

the deviation from the limit cycle, ∆a, and the angle on the

limit cycle, φ, is shown in Figure 4(a). It may be observed

that the map is approximately separable in the sense that

∆φb= b(φ)c(∆a)

(5)

where b and c are two different functions depending on

the sign of ∆a (inside or outside of the limit cycle). The

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comparison of the approximation versus the simulated shows

very small difference as indicated in Figure 4(b).

0

5

10

15

20

25

0

0.5

1

1.5

2

−1

−0.5

0

0.5

1

1.5

2

initial time (unit: hour)

Recover process Phase Lag from simulation

∆ a

steady state phase lag(unit: hour)

0

5

10

15

20

25

0

0.5

1

1.5

2

−1

−0.5

0

0.5

1

1.5

2

initial time (unit: hour)

Recover process Phase Lag from B(phi)*C(A)

∆ a

steady state phase lag(unit: hour)

Fig. 4.

a function of initial ∆a and time. (b) Phase lag map using separability

approximation

(a) Phase lag from recovery process based on simulation as

The describing function method is also useful in charac-

terizing effect of the light input on the limit cycle. The limit

cycle given approximated by the describing function is given

by

˙ z(ℓ)= (A+BBTη(1))z(ℓ)=

?

0

ω1

0

−ω2

?

z(ℓ)

(6)

where√ω1ω2= ω. The solution is

z(ℓ)=?

−ω1cos(ωt +ψ)

ωsin(ωt +ψ)

?

(7)

where ψ given by the initial condition. Linearize about this

approximate period orbit, and after removing the higher order

harmonics and higher order terms, the deviation from the

periodic orbit, δz := z−z(ℓ), is given approximately by

δ ˙ z =¯A(t)δz+¯B(t)u

(8)

¯A(t) = A+BBT(η(1)+g′(z(ℓ)(t))),¯B(t) = f1(z(ℓ)(t)). (9)

The approximation may be improved if z(ℓ)(t) in A(t) and

B(t) is replaced by z∗(t), the true periodic solution.

The solution of (8) is of the following general form

δz(t) = Σ(t,t0)δz(t0)+

?t

t0

Σ(t,τ)¯B(τ)u(τ)dτ

(10)

where Σ(t,t0) is the state transition matrix corresponding to

A(t). For an initial deviation ∆a from the orbit at angle φ,

the initial state is

δz(t0) =?

−cos(φ)

sin(φ)

?T∆a.

(11)

This shows the ensuing orbit and hence the phase is linear

in ∆a, explaining the separability property in (5). The time

varying linear equation (8) may also facilitate the computa-

tion of the phase response curve by using the discrete time

approximation of the differential equation.

In general, by examining the vector field, we can gain

some intuition of the effect of light input. As shown in

Figure 5, define etas the clockwise vector perpendicular to z

and enas the outward pointing vector perpendicular to f0(z).

We have the following characterizations:

Phase Change: f1(z)Tf0(z)

Amplitude Change: f1(z)Ten

Efficiency:

??f1(z)Tet

??/?z?.

Fig. 5.Definition of phase plane vectors enand et.

The calculation of efficiency is based on the approximation

that magnitude of the angular velocity,˙φ, is given by linear

tangential velocity divided by the radius.

By using these three characterizations, the effect of light

pulses in different portion of the state space is shown in

Figure 6. When the amplitude is small, i.e., closer to the

origin, the phase change efficiency is high. Therefore, a

judicious combination of the light pulses in different regimes

will achieve faster phase change. For example, one may want

to suppress amplitude by using light pulses in the amplitude

suppression region, change the phase in the high efficiency

region, and then regain the amplitude.

Fig. 6.

different portions of the phase plane

Effect of a light pulse on amplitude, phase, and efficiency in

C. Phase Control with Reference Pacemaker

An alternate approach is to use lighting control to drive the

circadian rhythm towards a reference oscillator (pacemaker).

Let zr denote the state of the reference oscillator with the

desired phase. Since it is simply the free running oscillator

with a different initial condition, it satisfies the unforced

dynamics:

˙ zr= f0(zr), zr(0) = zro.

(12)

Since the goal is to drive z towards zrby using u, we define

a standard error function (a Lyapunov function candidate):

V =1

2?z−zr?2.

(13)

The derivative of V along the trajectory is

˙V

=

=

(z−zr)T(f0(z)− f0(zr))+(z−zr)Tf1(z)u

(z−zr)TA(z−zr)+(z−zr)TB(g(BTz)−g(BTzr))

+(z−zr)Tf1(z)u.

To make˙V more negative, we can choose u to be

u = −u0sgn(min{fT

1(z)(z−zr),0})

(14)

which states that if the light input can reduce the tracking

error, then we turn on the light with intensity u0. Otherwise,

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the light is kept off. Then we have

˙V

=(z−zr)TA(z−zr)

?

+(−u0

?

?? ?

˙V1

+(z−zr)TB(g(BTz)−g(BTzr))

?

??min{fT

˙V3

??

.

?

˙V2

1(z)(z−zr),0}??)

???

At the present, we do not have a proof for the stability of the

closed loop system, and only have simulation results showing

convergence for various test scenarios. The stable behavior

that we have observed in simulation may be justified by the

following heuristic argument. Recall that A is unstable (one

of the eigenvalues of A+ATis negative, and the other is

close to zero), so˙V1is almost always positive. The function

g is incrementally negative when its argument is larger than

0.82. Therefore, if the error BT(z−zr) = x−xris large, the

second term becomes negative. The third term is negative

when the vectors f1(z) and z−zr form an obtuse angle.

Figure 7 shows the control of x with 6-hour initial phase

shift. The phase difference is almost completely removed in

20 hours with a single light pulse. Plotting the three terms of

˙V, as shown in Figure 8, illustrates the closed-loop behavior.

When the initial error is large, the (x −xr)(g(x) −g(xr))

term is large and negative, driving z−zr towards 0. When

?z−zr? is small, the negative term due to the feedback

action dominates, driving z−zrfurther towards zero. During

the period where lighting is off, ˙V does become positive,

and the tracking error increases. However, since the free

running periodic orbit is stable, the increase is bounded.

As the lighting control portion continues to reduce ?z−zr?,

the overall tracking error converges to zero. The plot of V

in Figure 9 shows that the convergence is not monotonic.

For the low light case, the convergence is slower but is still

towards zero.

Note that in the high lighting case, the amplitude is first

reduced. Once the phase difference is reduced, the amplitude

then increases back to the full magnitude.

0204060 80100

−1

−0.5

0

0.5

1

time (hr)

u0=0.5

controlled

reference

light input, uhat

−1.5−1 −0.50

x

0.511.5

−1.5

−1

−0.5

0

0.5

1

1.5

xc

u0=0.5

reference

controlled

start

Fig. 7.Phase compensation with trajectory error feedback, um= 0.5

Figure 10 considers another example. Using the proposed

control law, a 12-hour jet lag recovery is shortened to 3.5

days. Figure 11 plots the entrainment time versus the initial

phase difference. It is surprising to note that the 12-hour

entrainment time is 1 day shorter than the 7-hour entrainment

time! In fact, the 10-hour entrainment cost shows a local

minimum. This result appears to be counter-intuitive as

one may expect the entrainment time to be monotonically

increasing with respect to the initial phase difference. The

explanation of this phenomenon may be seen from the

01020304050

−0.16

−0.14

−0.12

−0.1

−0.08

−0.06

−0.04

−0.02

0

0.02

time (hr)

u0=0.5

vdot1

vdot2

vdot3

0 1020 304050

−0.3

−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

vdot

time (hr)

u0=0.5

Fig. 8.

overall˙V, um= 0.5

Comparison of the contribution of the three terms in˙V and the

0 10203040 50

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

V

time (hr)

0 100200 300400

time (hr)

500 600700800900

1.6

1.7

1.8

1.9

2

2.1

2.2

2.3

V

u0=0.01

Fig. 9.Comparison of V for um= 0.5 and um= 0.01

trajectory in the phase plane. In Figure 12, the trajectory of

the 7-hour case winds around the origin while the trajectory

of the 10-hour case passes almost through the center. From

the efficiency analysis and the phase response map shown in

Figure 6, we observe that near the origin of phase plane, both

phase advancing efficiency and phase delaying efficiency are

much larger. Hence, the 10-hour trajectory can catch up with

the reference trajectory more quickly.

01234567

−1.5

−1

−0.5

0

0.5

1

1.5

2

time (day)

controlled

reference

light input

Fig. 10.

the 10,000lux light stimulus

12-hour jet lag recovery with phase plane tracking. Red line is

−10−50510

0

1

2

3

4

Initial phase difference (hr)

Entrainment time cost (days)

Fig. 11. entrainment time cost versus initial phase difference

D. A Two-Step Approach to Phase Control

The observation of reducing the amplitude to gain effi-

ciency motivates a modified two-step strategy: first drive the

oscillator into the center and then let it catch up with the

reference trajectory. To drive the oscillator to the center, we

apply light when the oscillator is in the suppressed-amplitude

region. However, from simulation and experiments, driving

a oscillator into the center region takes about 3 days [18],

which adds to the entrainment time . Hence, we first drive the

trajectory toward the center until it comes within a predefined

Page 6

Fig. 12.

phase shift. The 10-hour phase shift trajectory passes near the origin where

the phase shifting efficiency is high.

The phase plane trajectories for the (a) 7-hour and (b) 10-hour

distance R from the origin. The control law (14) is then

applied to match up the phase. The optimal distance R is

may be be found for each initial phase difference using

nonlinear optimization. It can then be encoded in a look-

up table. Figure 13 shows the entrainment time versus initial

phase difference with this modified method, demonstrating a

significant reduction of the entrainment time.

−10−505 10

0

0.5

1

1.5

2

2.5

3

3.5

4

Initial phase difference (hours)

Entrainment time cost (days)

Phase plane tracking

Improved Phase plane tracking

Fig. 13. With improved phase plane tracking method, the entrainment time

cost is shortened by up to 1.5 days.

V. CONCLUSION AND FUTURE WORK

This paper analyzed an empirical circadian rhythm model

developed by Kronauer. We showed that in the free running

case (no light), the describing function method gives excel-

lent prediction of the oscillation. The linearization about this

approximate periodic solution has the potential of being used

for forced oscillation analysis as well. By using the nonlinear

empirical model, we show two approaches to shift the phase

of the oscillation. The first approach is similar to the PRC

method, but with a more detailed analysis of the impact of a

pulse at different portions of the orbit. The second approach

is based on a reference oscillation, and the feedback of the

state error. A Lyapunov-like analysis is used to heuristically

argue that the system is stable. Simulation shows that the

approach has excellent transient performance, reducing jet

lag recovery from 7 days to 2.5 days.

We are currently working with drosophila experiments

to compare the biochemical model [3] and the empirical

oscillator model [8] by using the activity measurements as

an indirect indicator of the circadian rhythm. The experi-

mentally identified model will be used for the estimation

of circadian rhythm based on measured activity. We are

also investigating the spectral response of the circadian

system [19]and the use of spectral-tunable lighting to balance

between circadian system control and lighting needed for

visual perception.

ACKNOWLEDGMENT

The authors would like to thank Mark Rea for his help

and collaboration in this research. The authors would also

like to thank Professor Bernard Possidente for his generous

guidance and support in drosophila experiment. This work

was supported in part by the National Science Foundation

(NSF) Smart Lighting Engineering Research Center (EEC-

0812056), in part by the Center for Automation Technologies

and Systems (CATS) under a block grant from the New York

State Foundation for Science, Technology and Innovation

(NYSTAR), and in part by the Lighting Research Center

(LRC).

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