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Explaining the East/West Asymmetry of Jet Lag

Dimension Reduction Methods for Analyzing Networks of Circadian Oscillators

By Thomas M. Antonsen, Michelle Girvan, Zhixin Lu, and Edward Ott

Jet lag is a common experience for airplane travelers crossing multiple time zones. Typical symptoms include drowsiness, discomfort, reduced functionality during the local daytime, and difficulty sleeping during the local nighttime. Simply explained, the human body follows a circadian rhythm that synchronizes with the local 24-hour day/night cycle of external natural conditions (particularly the rising and setting of the sun) and social conditions. Upon rapid crossing of several time zones, the body’s circadian oscillation needs time to resynchronize to the local oscillation phase of the external conditions; this resynchronization phase manifests as jet lag symptoms in travelers. Since resynchronization of an oscillator is a dynamical process, this phenomenon lends itself to mathematical modeling from a dynamical systems perspective.

While the body produces many signals that help determine its circadian rhythm, one bodily region seems particularly important in this process: the suprachiasmatic nucleus (SCN), a tiny region of the brain’s hippocampus. Physiological studies show that the SCN contains of the order of \(10^4\) neural oscillators, and that it is reasonable to assume that, in isolation, the periods of individual oscillators are distributed with a small dispersion around a mean slightly longer than 24 hours. When coupled together within the SCN, these oscillators are thought to undergo collective synchronization with each other as well as with external stimuli experienced by the individual, e.g., the rising and setting of the sun. This information provides the basis for our model’s construction, as well as others that have preceded it. The modeling that we employ [4], however, is different from previous attempts in that we start at the microscopic level of the individual coupled SCN oscillators, but then reduce the high-dimensional microscopic description to a low-dimensional macroscopic description. Our model’s primary purpose is to address the empirical observation that jet lag is more severe (i.e., requires a longer recovery time) for eastward travel than for westward travel across the same number of time zones. Specifically, we explore to what extent this east/west jet lag asymmetry may be explained by the small amount by which the average SCN period exceeds 24 hours.

Since we wish to understand the interplay between global travel and resynchronization in a large collection of \(N\) neuronal oscillators, we use a very simplified model for the neuronal oscillators, perhaps at some expense of realism. As in the well-known Kuramoto model [1], the complicated dynamics of each oscillator are reduced to a time evolution equation for a phase \(\theta_i(t)\), representing the state of the \(i^{th}\) oscillator where \(i=1,2,...,N.\) The phases of the oscillators advance in time, according to the model

\[ \frac{d\theta_i}{dt} = \omega_i + \frac{K}{N} \sum_{i=1}^{N} sin (\theta_j - \theta_i) + F sin(\sigma t - \theta_i + p). \tag1 \]

Here, \(\omega_i\) is the natural frequency for each oscillator which, in our model, is drawn from a distribution \(g(\omega_i)\) that peaks at a value of \(\omega\), corresponding to a period just over 24 hours. The second term on the right describes the cells’ tendency to synchronize with each other, and has coupling strength \(K\). The third term represents interaction with the outside world, particularly the effect of sunlight. It attracts, with strength \(F\), each cell to a phase \(\sigma t + p\), where \(\sigma = 2 \pi/24\: \textrm{hrs}^{-1}\) is the daily frequency and \(p\) is a phase that depends on time zone. If \(t\) is Greenwich Mean Time, positive \(p\),  \(0 < p < \pi\), corresponds to east of Greenwich and negative \(p\),  \(0 > p > -\pi\), to west of Greenwich.

As simple as this model is, it still requires solving a large number, \(N \sim 10^4\), of coupled equations. We address this requirement with two further simplifications. First, we pass to the continuum limit, \(N \to \infty\), where the state of the SCN is characterized by a time-dependent distribution of oscillator phases and frequencies, \(f(\theta, \omega, t)\). This \(N \to \infty\) limit should be a good approximation for \(N \gg 1\). Second, we use the so-called Ott-Antonsen ansatz [5], which represents an exact solution for the distribution \(f\). This ansatz postulates a particular form for \(f(\theta, \omega, t)\), which yields a reduced system when substituted into the continuum system. Furthermore, [6] shows that under weak conditions and at large time, \(f\) converges in probability to the solution of the reduced description. Thus, we capture all attractors and bifurcations. In the case of a Lorenzian distribution of natural oscillator frequencies, \(g(\omega) = (\triangle / \pi)\lbrack(\omega - \omega_0)^2 + \triangle^2\rbrack^{-1}\), the macroscopic state of the SCN is described by a single complex variable \(z=N^{-1}\sum_{j}exp \lbrack i(\theta_j - \sigma t - p)\rbrack\), which evolves according to the equation

\[\frac{dz(t)}{dt} = \frac{1}{2} \lbrack (Kz + F) - (Kz + F)^* z^2\rbrack - \big(\triangle + i(\omega_0 - \sigma)\big) z, \tag2 \] where the polar angle of the complex variable \(z\) represents the collective global oscillation phase of the SCN. Thus, the ansatz reduces an \(N\)-dimensional system to this single, first-order, complex ordinary differential equation, enabling rapid scanning in parameter space and enhanced understanding of the dynamics.

Although the ansatz owes its discovery to an investigation [2] into the macroscopic behavior of solutions of \((1)\), it turns out that this dimension reduction result applies, not only to \((1)\) [3] but to a very large class of interesting situations. These include, for example, a model of pedestrian-induced wobbling of London’s Millennium Bridge, Josephson junction circuits, a model of birdsong, and networks of pulse-coupled neurons. One can also generalize it to include additional dynamical features, like time delays in the effect of one oscillator upon another, the effects of different types of network topology, spatial coupling, and feedback control.

The model \((2)\) has three parameter regimes with qualitatively different dynamics, as depicted in Figure 1. We expect a healthy person’s circadian rhythm to entrain with the external 24-hour period, which corresponds to a stable fixed point in the \(z\)-phase-space (the black dot in Figure 1a and 1b). An individual whose circadian rhythm is not synched with the external 24-hour period corresponds to a \(z\)-phase-portrait, as shown in Figure 1c, where the individual’s circadian phase relative to the phase of the external drive continually drifts around a closed curve, a periodic orbit in \(z\). There is a difference between dynamics corresponding to Figure 1a and 1b; through a saddle-node bifurcation, Figure 1b has—besides the stable fixed point—two other fixed points, one unstable (shown as an open circle) and one a saddle (shown as a cross). The unstable manifold of the saddle forms a loop, along which \(z\) can approach the stable fixed point (black dot) from two opposite directions.

Figure 1. Trajectories of z(t) from (2) for three different parameter sets. Image Credit: [4].

To analyze recovery of a healthy individual from jet lag, we assume that the traveler is entrained to his/her pre-travel time zone (\(z\) at the stable fixed point) before the trip. For simplicity, we also assume that the traveler’s cross-time-zone travel is very fast, and model it as a discontinuous change of \(p\) in \((1)\). Thus, immediately after travel, the state variable \(z\) is suddenly displaced by a rotation \((|z| \textrm{fixed})\) by the angle \(\lbrack p(initial) - p(final)\rbrack\). Depending on where the trip ends, \(z\) moves back to the stable fixed point either by advancing or delaying its phase. We are particularly interested in the east-west asymmetry in the direction of recovery and the time it takes for the recovery to occur. We use a ‘typical’ set of parameters representative of a typical healthy individual. This set of parameters yields the dynamics shown in Figure 1a, where only one fixed point is stable. The mean oscillation period of SCN cells when external drive is absent is taken to be 24.5 hours, consistent with experimental observations. This computation surprisingly indicates that the small amount (~30 minutes) by which the natural SCN period exceeds 24 hours in a typical human is sufficient to explain the rather noticeable east-west asymmetry of jet lag.

Edward Ott will present the Jürgen Moser Lecture, “Emergent Behavior in Large Systems of Many Coupled Oscillators,” at the SIAM Conference on Applications of Dynamical Systems (DS17), to be held in Snowbird, UT, this May. He will also organize and give a talk at a minisymposium on “Using reservoir computers to learn dynamical systems,” while Michelle Girvan will speak at a minisymposium titled “Symmetry, Asymmetry, and Network Synchronization.” 

References
[1] Acebrón, J.A., Bonilla, L.L., Vicente, C.J.P., Ritort, F., & Spigler, R. (2005). The Kuramoto model: A simple paradigm for synchronization phenomena. Reviews of Modern Physics, 77(1), 137.
[2] Antonsen Jr., T.M., Faghih, R.T., Girvan, M., Ott, E., & Platig, J. (2008). External periodic driving of large systems of globally coupled phase oscillators. Chaos: An Interdisciplinary Journal of Nonlinear Science, 18(3), 037112.
[3] Childs, L.M., & Strogatz, S.H. (2008). Stability diagram for the forced Kuramoto model. Chaos: An Interdisciplinary Journal of Nonlinear Science, 18(4), 043128.
[4] Lu, Z., Klein-Cardeña, K., Lee, S., Antonsen, T.M., Girvan, M., & Ott, E. (2016). Resynchronization of circadian oscillators and the east-west asymmetry of jet-lag. Chaos: An Interdisciplinary Journal of Nonlinear Science, 26(9), 094811.
[5] Ott, E., & Antonsen, T.M. (2008). Low dimensional behavior of large systems of globally coupled oscillators. Chaos: An Interdisciplinary Journal of Nonlinear Science, 18(3), 037113.
[6] Ott, E., & Antonsen, T.M. (2009). Long time evolution of phase oscillator systems. Chaos: An Interdisciplinary Journal of Nonlinear Science, 19(2), 023117.

The authors are all affiliated with the Institute for Research in Electronics and Applied Physics at the University of Maryland, College Park. Additionally, Thomas Antonsen and Edward Ott hold joint appointments in the Department of Physics and the Department of Electrical and Computer Engineering. Michelle Girvan holds appointments in the Department of Physics and the Institute for Physical Science and Technology. Zhixin Lu is a research graduate student in the Chemical Physics Program.

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