References Conclusions

Resynchroniza6on%of%Circadian%Oscillators%and%the% East:West%Asymmetry%of%Jet:Lag%% Zhixin Lua, Kevin Klein-Cardeñab, Steven Leec, Thomas Antonsena, Michelle Girvana, and Edward Otta aUniversity of Maryland, College Park, MD 20742, bDePaul University, Chicago, IL 60604, cBrooklyn College, Brooklyn, NY 11210

Introduction

Phase Diagrams

Circadian rhythms in mammals are regulated by suprachiamatic nucleus (SCN) which consists of ~104 synchronized pacemaker neuronal cells.

Type A dynamics

Type B dynamics

Type C dynamics

SCN

zst

zst entrainable%%

non:entrainable%%

entrainable%%

Light induced neuronal signals

24-hour-period light-dark cycle

Figs. 3: Dynamics in z-space. Type A has only one stable fixed point. Type B has an attracting limit cycle and an unstable fixed pt. Type C has one stable, one unstable, and one saddle fixed point.

Fig. 1: Human circadian system

Distribution of ωi

natural% frequency%

✓i ) + F sin( t

coupling%from% other%neurons%

g(!) =

1

1

!

!0

Fig. 2: Natural frequency distribution We adopt Greenwich Mean Time for time t and assign each time zone a phase value p within the range [-π,π). We set p=0 at Greenwich, p>0 eastward, and p ⌧

[phase%@%des6na6on]%

Order Parameter Dynamics z

order parameter defined in a rotating frame: % N X 1 z= ei[✓j (t) t p(t)] N j=1

Using Ott-Antonsen ansatz, we obtain the dynamical equation before/after the travel: ⇤ 1⇥ 2 ⇤ z˙ = (Kz + F ) z (Kz + F ) ( + i⌦)z 2 At the moment of the instantaneous travel: z(⌧ + ) = z(⌧ )e

zst | Fig. 5: Jet-lag recovery over time (for Type A dynamics). Fully recovery threshold is ~0.2.

2]

2⇡ = (hr) 24 2⇡ !0 ⇡ (hr) 25

⌦

✓j

days%a&er%travel%

24hr%periodic%external% driving%@%6me:zone%p"

!0 ) 2 +

⇡[(!

✓i + p(t))

i(p2 p1 )

Type A dynamics

N X d✓i K = !i + sin(✓j dt N j=1

East-West Asymmetry

|z(t)

Phase Oscillator Model

Resynchronization

Type C dynamics

We start with a high dimensional microscopic description of the dynamics of individual SCN cells and then use an analytical dimension reduction method[1] to obtain a low dimensional, macroscopic description. Using this we study the re-entrainment dynamics and explain why jet-lag after eastward travel is tougher.

⌦/

Figs. 4: Re-entrainment after cross-time-zonetravels (5.5hr(E), 6.5hr(E), 12hr(E/W), and 6hr(W)). Type B dynamics is not entrainable and thus is not shown.

Stable manifold of the saddle point in Fig. 4(a) between 5.5hr(E) and 6.5hr(E) is a separatrix of phase-advancing/delaying re-entrainments. The recovery from 6hr(E)-jet-lag is the toughest due to the stable manifold. By decreasing K, the saddle point and unstable fixed point in Fig. 4(a) disappear through a saddle-node bifurcation that leads to Fig. 4(b), yet similar properties remain.

Fig. 6: Recovery time with positive/negative Ω values. When positive, eastward jet-lag is tougher than westward and vice versa.

Conclusions •  We employ forced Kuramoto model to study the re-entrainment of circadian rhythms. •  Our model shows the experimentally reported ‘wrong direction’ re-entrainment after large enough eastward travels. •  East-west asymmetry of severity of jet-lag is also explained.

References [1] E. Ott and T. M. Antonsen, Chaos 18, 037113 (2008). [2] T. Antonsen Jr, R. Faghih, M. Girvan, E. Ott, and J. Platig, Chaos 18, 037112 (2008) [3] Y. Yamaguchi, et al., Science 342, 85 (2013).

We gratefully acknowledge financial support from NSF under grant PHY-1461089, and ARO under contract number W911NF-12-1-0101.