Resynchronization of circadian oscillators and the east-west asymmetry of jet-lag

The seasons within: a theoretical perspective on photoperiodic entrainment and encoding

Circadian clocks are internal timing devices that have evolved as an adaption to the omnipresent natural 24 h rhythmicity of daylight intensity. Properties of the circadian system are photoperiod dependent. The phase of entrainment varies systematically with season. Plastic photoperiod-dependent re-arrangements in the mammalian circadian core pacemaker yield an internal representation of season. Output pathways of the circadian clock regulate photoperiodic responses such as flowering time in plants or hibernation in mammals. Here, we review the concepts of seasonal entrainment and photoperiodic encoding. We introduce conceptual phase oscillator models as their high level of abstraction, but, yet, intuitive interpretation of underlying parameters allows for a straightforward analysis of principles that determine entrainment characteristics. Results from this class of models are related and discussed in the context of more complex conceptual amplitude-phase oscillators as well as contextual molecular models that take into account organism, tissue, and cell-type-specific details.

A minimal model of peripheral clocks reveals differential circadian re-entrainment in aging

The mammalian circadian system comprises a network of endogenous oscillators, spanning from the central clock in the brain to peripheral clocks in other organs. These clocks are tightly coordinated to orchestrate rhythmic physiological and behavioral functions. Dysregulation of these rhythms is a hallmark of aging, yet it remains unclear how age-related changes lead to more easily disrupted circadian rhythms. Using a two-population model of coupled oscillators that integrates the central clock and the peripheral clocks, we derive simple mean-field equations that can capture many aspects of the rich behavior found in the mammalian circadian system. We focus on three age-associated effects that have been posited to contribute to circadian misalignment: attenuated input from the sympathetic pathway, reduced responsiveness to light, and a decline in the expression of neurotransmitters. We find that the first two factors can significantly impede re-entrainment of the clocks following perturbation, while a weaker coupling within the central clock does not affect the recovery rate. Moreover, using our minimal model, we demonstrate the potential of using the feed-fast cycle as an effective intervention to accelerate circadian re-entrainment. These results highlight the importance of peripheral clocks in regulating the circadian rhythm and provide fresh insights into the complex interplay between aging and the resilience of the circadian system.

Multistability in coupled oscillator systems with higher-order interactions and community structure

We study synchronization dynamics in populations of coupled phase oscillators with higher-order interactions and community structure. We find that the combination of these two properties gives rise to a number of states unsupported by either higher-order interactions or community structure alone, including synchronized states with communities organized into clusters in-phase, anti-phase, and a novel skew-phase, as well as an incoherent-synchronized state. Moreover, the system displays strong multistability with many of these states stable at the same time. We demonstrate our findings by deriving the low dimensional dynamics of the system and examining the system's bifurcations using stability analysis and perturbation theory.

Limits of entrainment of circadian neuronal networks

Circadian rhythmicity lies at the center of various important physiological and behavioral processes in mammals, such as sleep, metabolism, homeostasis, mood changes, and more. Misalignment of intrinsic neuronal oscillations with the external day-night cycle can disrupt such processes and lead to numerous disorders. In this work, we computationally determine the limits of circadian synchronization to external light signals of different frequency, duty cycle, and simulated amplitude. Instead of modeling circadian dynamics with generic oscillator models (e.g., Kuramoto-type), we use a detailed computational neuroscience model, which integrates biomolecular dynamics, neuronal electrophysiology, and network effects. This allows us to investigate the effect of small drug molecules, such as Longdaysin, and connect our results with experimental findings. To combat the high dimensionality of such a detailed model, we employ a matrix-free approach, while our entire algorithmic pipeline enables numerical continuation and construction of bifurcation diagrams using only direct simulation. We, thus, computationally explore the effect of heterogeneity in the circadian neuronal network, as well as the effect of the corrective therapeutic intervention of Longdaysin. Last, we employ unsupervised learning to construct a data-driven embedding space for representing neuronal heterogeneity.

Greedy optimization for growing spatially embedded oscillatory networks

The coupling of some types of oscillators requires the mediation of a physical link between them, rendering the distance between oscillators a critical factor to achieve synchronization. In this paper, we propose and explore a greedy algorithm to grow spatially embedded oscillator networks. The algorithm is constructed in such a way that nodes are sequentially added seeking to minimize the cost of the added links' length and optimize the linear stability of the growing network. We show that, for appropriate parameters, the stability of the resulting network, measured in terms of the dynamics of small perturbations and the correlation length of the disturbances, can be significantly improved with a minimal added length cost. In addition, we analyze numerically the topological properties of the resulting networks, and we find that, while being more stable, their degree distribution is approximately exponential and independent of the algorithm parameters. Moreover, we find that other topological parameters related with network resilience and efficiency are also affected by the proposed algorithm. Finally, we extend our findings to more general classes of networks with different sources of heterogeneity. Our results are a step in the development of algorithms for the directed growth of oscillatory networks with desirable stability, dynamical and topological properties.

Entrainment within hierarchical circadian oscillator networks

Circadian rhythms are endogenous oscillations, widely found across biological species, that have the capability of entraining to the 24-h light-dark cycle. Circadian systems often consist of both central oscillators that receive direct light-dark input and peripheral oscillators that receive input from the central oscillators. In this paper, we address questions related to what governs the time to and pattern of entrainment of these hierarchical circadian systems after an abrupt switch in the light-dark phasing. For a network consisting of a single central oscillator coupled to a chain of N feed-forward peripheral oscillators, we introduce a systematic way to derive an N-dimensional entrainment map whose fixed points correspond to entrained solutions. Using the map, we explain that the direction of reentrainment can involve fairly complicated phase advancing and delaying behavior as well as reentrainment times that depend sensitively on the nature of the perturbation. We also study the dynamics of a hierarchical system in which the peripheral oscillators are mutually coupled. We study how reentrainment times vary as a function of the degree to which the oscillators are desynchronized at the time of the change in light-dark phasing. We show that desynchronizing the peripheral oscillators can, in some circumstances, speed up their ultimate reentrainment following perturbations.

Generation and Disruption of Circadian Rhythms in the Suprachiasmatic Nucleus: A Core-Shell Model

We focus our research on how the core-shell organization controls behavior of the suprachiasmatic nucleus (SCN), how the core and shell are synchronized to the environment, what impact they have on the behavior of the SCN under different lighting conditions, and what mechanisms disrupt synchronization. To this end, we use a reduced Kuramoto model, with parameters inferred from experimental observations and calibrated for mice, and perform a detailed comparison between the model and experimental data under light-dark (LD), dark-dark (DD), and light-light (LL) conditions. The operating limits of free-running and entrained SCN activity under symmetric LD cycles are analyzed, with particular focus on the phenomena of anticipation and dissociation. Results reveal that the core-shell organization of the SCN enables anticipation of future events over circadian cycles. The model predicts the emergence of a second (dissociated) rhythm for large and small LD periods. Our results are in good qualitative and quantitative agreement with experimental observations of circadian dissociation. We further describe SCN activity under LL conditions and show that our model satisfies Aschoff’s first rule, according to which the endogenous free-running circadian period observed under complete darkness will shorten in diurnal animals and lengthen in nocturnal animals under constant light. Our results strongly suggest that the Kuramoto model captures essential features of synchronization and entrainment in the SCN. Moreover, our approach is easily extendible to an arbitrary number of groups, with dynamics described by explicit equations for the group phase and synchronization index. Viewed together, the reduced Kuramoto model presents itself as a useful tool for exploring open problems in the study of circadian rhythms, one that can account for evolving views of the circadian system’s organization, including peripheral clocks and inter-hemispheric interaction, and can be translated to other nocturnal and diurnal animals, including humans.

Optimization of light exposure and sleep schedule for circadian rhythm entrainment

The circadian rhythm, called Process C, regulates a wide range of biological processes in humans including sleep, metabolism, body temperature, and hormone secretion. Light is the dominant synchronizer of the circadian rhythm-it has been used to regulate the circadian phase to cope with jet-lag, shift work, and sleep disorder. The homeostatic oscillation of the sleep drive is called Process S. Process C and Process S together determine the sleep-wake cycle in what is known as the two-process model. This paper addresses the regulation of both Process C and Process S by scheduling light exposure and sleep based on numerical simulations of circadian rhythm and sleep mathematical models. This is a significant step beyond the existing literature that only considers the entrainment of Process C. Regulation of the two-process model poses several unique features and challenges: 1. Process S is non-smooth, i.e., the homeostatic dynamics are different in the sleep and wake regimes; 2. Light only indirectly affects Process S through Process C; 3. Light does not affect Process C during sleep. We consider two scenarios: optimizing light intensity as the control input with spontaneous (i.e., unscheduled) sleep/wake times and jointly optimizing the light intensity and the sleep/wake times, which allows limited delayed sleep and early waking as part of the decision variables. We solve the time-optimal entrainment problem for the two-process model for both scenarios using an extension of the gradient descent algorithm to non-smooth systems. To illustrate the efficacy of our time-optimal entrainment strategies, we consider two common use cases: transmeridian travelers and shift workers. For transmeridian travelers, joint optimization of the two-process model avoids the unrealistic long wake duration when only Process C is considered. The entrainment time also decreases when both the light input and the sleep schedule are optimized compared to when only the light input is optimized. For shift workers, we show that the entrainment time is significantly shortened by optimizing the night shift working light.

The lower bound of the network connectivity guaranteeing in-phase synchronization

In-phase synchronization is a stable state of identical Kuramoto oscillators coupled on a network with identical positive connections, regardless of network topology. However, this fact does not mean that the networks always synchronize in-phase because other attractors besides the stable state may exist. The critical connectivity μ_c is defined as the network connectivity above which only the in-phase state is stable for all the networks. In other words, below μ_c, one can find at least one network that has a stable state besides the in-phase sync. The best known evaluation of the value so far is 0.6828…≤μ_c≤0.7889. In this paper, focusing on the twisted states of the circulant networks, we provide a method to systematically analyze the linear stability of all possible twisted states on all possible circulant networks. This method using integer programming enables us to find the densest circulant network having a stable twisted state besides the in-phase sync, which breaks a record of the lower bound of the μ_c from 0.6828… to 0.6838…. We confirm the validity of the theory by numerical simulations of the networks not converging to the in-phase state.

Delays are Self-enhancing: An Explanation of the East-West Asymmetry in Recovery from Jetlag

There is evidence in mammals that recovering from jetlag after westward travel is faster than after eastward travel. To understand why, mathematical models have been used, along with theories of entrainment rooted in experimental evidence. The most complete understanding relies on detailed mathematical modeling, so it is helpful to develop an intuition about why there is an east-west asymmetry. One such intuition is that humans have long periods and therefore recover better when they can delay. Although this is part of the reason, it does not explain why short-period mice also recover from westward travel faster. Our goal is to provide a simple intuition consistent with detailed mathematical theories, but which does not require mathematical expertise to follow. Here, we present the intuition that westward travel is easier to recover from because of a simple principle: delays are self-enhancing.

Seasonality and light phase-resetting in the mammalian circadian rhythm

We study the impact of light on the mammalian circadian system using the theory of phase response curves. Using a recently developed ansatz we derive a low-dimensional macroscopic model for the core circadian clock in mammals. Significantly, the variables and parameters in our model have physiological interpretations and may be compared with experimental results. We focus on the effect of four key factors which help shape the mammalian phase response to light: heterogeneity in the population of oscillators, the structure of the typical light phase response curve, the fraction of oscillators which receive direct light input and changes in the coupling strengths associated with seasonal day-lengths. We find these factors can explain several experimental results and provide insight into the processing of light information in the mammalian circadian system. In particular, we find that the sensitivity of the circadian system to light may be modulated by changes in the relative coupling forces between the light sensing and non-sensing populations. Finally, we show how seasonal day-length, after-effects to light entrainment and seasonal variations in light sensitivity in the mammalian circadian clock are interrelated.

Conceptual Models of Entrainment, Jet Lag, and Seasonality

Understanding entrainment of circadian rhythms is a central goal of chronobiology. Many factors, such as period, amplitude, Zeitgeber strength, and daylength, govern entrainment ranges and phases of entrainment. We have tested whether simple amplitude-phase models can provide insight into the control of entrainment phases. Using global optimization, we derived conceptual models with just three free parameters (period, amplitude, and relaxation rate) that reproduce known phenotypic features of vertebrate clocks: phase response curves (PRCs) with relatively small phase shifts, fast re-entrainment after jet lag, and seasonal variability to track light onset or offset. Since optimization found multiple sets of model parameters, we could study this model ensemble to gain insight into the underlying design principles. We found complex associations between model parameters and entrainment features. Arnold onions of representative models visualize strong dependencies of entrainment on periods, relative Zeitgeber strength, and photoperiods. Our results support the use of oscillator theory as a framework for understanding the entrainment of circadian clocks.

Phase-amplitude reduction far beyond the weakly perturbed paradigm

While phase reduction is a well-established technique for the analysis of perturbed limit cycle oscillators, practical application requires perturbations to be sufficiently weak thereby limiting its utility in many situations. Here, a general strategy is developed for constructing a set of phase-amplitude reduced equations that is valid to arbitrary orders of accuracy in the amplitude coordinates. This reduction framework can be used to investigate the behavior of oscillatory dynamical systems far beyond the weakly perturbed paradigm. Additionally, a patchwork phase-amplitude reduction method is suggested that is useful when exceedingly large magnitude perturbations are considered. This patchwork method incorporates the high-accuracy phase-amplitude reductions of multiple nearby periodic orbits that result from modifications to nominal parameters. The proposed method of high-accuracy phase-amplitude reduction can be readily implemented numerically and examples are provided where reductions are computed up to fourteenth order accuracy.

Circadian rhythm disruption and mental health

Circadian rhythms are internal manifestations of the solar day that permit adaptations to predictable environmental temporal changes. These ~24-h rhythms are controlled by molecular clockworks within the brain that are reset daily to precisely 24 h by exposure to the light-dark cycle. Information from the master clock in the mammalian hypothalamus conveys temporal information to the entire body via humoral and neural communication. A bidirectional relationship exists between mood disorders and circadian rhythms. Mood disorders are often associated with disrupted circadian clock-controlled responses, such as sleep and cortisol secretion, whereas disruption of circadian rhythms via jet lag, night-shift work, or exposure to artificial light at night, can precipitate or exacerbate affective symptoms in susceptible individuals. Evidence suggests strong associations between circadian rhythms and mental health, but only recently have studies begun to discover the direct interactions between the circadian system and mood regulation. This review provides an overview of disrupted circadian rhythms and the relationship to behavioral health and psychiatry. The focus of this review is delineating the role of disruption of circadian rhythms on mood disorders using human night shift studies, as well as jet lag studies to identify links. We also review animal models of disrupted circadian rhythms on affective responses. Lastly, we propose low-cost behavioral and lifestyle changes to improve circadian rhythms and presumably behavioral health.

Dynamical equivalence between Kuramoto models with first- and higher-order coupling

The Kuramoto model with high-order coupling has recently attracted some attention in the field of coupled oscillators in order, for instance, to describe clustering phenomena in sets of coupled agents. Instead of considering interactions given directly by the sine of oscillators' angle differences, the interaction is given by the sum of sines of integer multiples of these angle differences. This can be interpreted as a Fourier decomposition of a general 2π-periodic interaction function. We show that in the case where only one multiple of the angle differences is considered, which we refer to as the "Kuramoto model with simple qth-order coupling," the system is dynamically equivalent to the original Kuramoto model. In other words, any property of the Kuramoto model with simple higher-order coupling can be recovered from the standard Kuramoto model.

Macroscopic Models for Human Circadian Rhythms

Mathematical models have a long and influential history in the study of human circadian rhythms. Accurate predictive models for the human circadian light response have been used to study the impact of a host of light exposures on the circadian system. However, generally, these models do not account for the physiological basis of these rhythms. We illustrate a new paradigm for deriving models of the human circadian light response. Beginning from a high-dimensional model of the circadian neural network, we systematically derive low-dimensional models using an approach motivated by experimental measurements of circadian neurons. This systematic reduction allows for the variables and parameters of the derived model to be interpreted in a physiological context. We fit and validate the resulting models to a library of experimental measurements. Finally, we compare model predictions for experimental measurements of light levels and discuss the differences between our model’s predictions and previous models. Our modeling paradigm allows for the integration of experimental measurements across the single-cell, tissue, and behavioral scales, thereby enabling the development of accurate low-dimensional models for human circadian rhythms.

The central role of peripheral nodes in directed network dynamics

Many social, technological, and biological systems with asymmetric interactions display a variety of collective phenomena, such as opinion formation and synchronization. This has motivated much research on the dynamical impact of local and mesoscopic structure in directed networks. However, the unique constraints imposed by the global organization of directed networks remain largely undiscussed. Here, we control the global organization of directed Erdős–Rényi networks, and study its impact on the emergence of synchronization and ferromagnetic ordering, using Kuramoto and Ising dynamics. In doing so, we demonstrate that source nodes – peripheral nodes without incoming links – can disrupt or entirely suppress the emergence of collective states in directed networks. This effect is imposed by the bow-tie organization of directed networks, where a large connected core does not uniquely ensure the emergence of collective states, as it does for undirected networks.

Optimal global synchronization of partially forced Kuramoto oscillators

We consider the problem of global synchronization in a large random network of Kuramoto oscillators where some of them are subject to an external periodically driven force. We explore a recently proposed dimensional reduction approach and introduce an effective two-dimensional description for the problem. From the dimensionally reduced model, we obtain analytical predictions for some critical parameters necessary for the onset of a globally synchronized state in the system. Moreover, the low dimensional model also allows us to introduce an optimization scheme for the problem. Our main conclusion, which has been corroborated by exhaustive numerical simulations, is that for a given large random network of Kuramoto oscillators, with random natural frequencies ω_i, such that a fraction of them is subject to an external periodic force with frequency Ω, the best global synchronization properties correspond to the case where the fraction of the forced oscillators is chosen to be those ones such that |ω_i-Ω| is maximal. Our results might shed some light on the structure and evolution of natural systems for which the presence or the absence of global synchronization is a desired property. Some properties of the optimal forced networks and their relation to recent results in the literature are also discussed.

Observing microscopic transitions from macroscopic bursts: Instability-mediated resetting in the incoherent regime of the D-dimensional generalized Kuramoto model

This paper considers a recently introduced D-dimensional generalized Kuramoto model for many (N≫1) interacting agents, in which the agent states are D-dimensional unit vectors. It was previously shown that, for even (but not odd) D, similar to the original Kuramoto model (D=2), there exists a continuous dynamical phase transition from incoherence to coherence of the time asymptotic attracting state (time t→∞) as the coupling parameter K increases through a critical value which we denote K_c ⁽⁺⁾>0. We consider this transition from the point of view of the stability of an incoherent state, where an incoherent state is defined as one for which the N→∞ distribution function is time-independent and the macroscopic order parameter is zero. In contrast with D=2, for even D>2, there is an infinity of possible incoherent equilibria, each of which becomes unstable with increasing K at a different point K=K_c. Although there are incoherent equilibria for which K_c=K_c ⁽⁺⁾, there are also incoherent equilibria with a range of possible K_c values below K_c ⁽⁺⁾, (K_c ⁽⁺⁾/2)≤K_c<K_c ⁽⁺⁾. How can the possible instability of incoherent states arising at K=K_c<K_c ⁽⁺⁾ be reconciled with the previous finding that, at large time (t→∞), the state is always incoherent unless K>K_c ⁽⁺⁾? We find, for a given incoherent equilibrium, that, if K is rapidly increased from K<K_c to K_c<K<K_c ⁽⁺⁾, due to the instability, a short, macroscopic burst of coherence is observed, in which the coherence initially grows exponentially, but then reaches a maximum, past which it decays back into incoherence. Furthermore, after this decay, we observe that the equilibrium has been reset to a new equilibrium whose K_c value exceeds that of the increased K. Thus, this process, which we call "Instability-Mediated Resetting," leads to an increase in the effective K_c with continuously increasing K, until the equilibrium has been effectively set to one for which K_c≈K_c ⁽⁺⁾. Thus, instability-mediated resetting leads to a unique critical point of the t→∞ time asymptotic state (K=K_c ⁽⁺⁾) in spite of the existence of an infinity of possible pretransition incoherent states.

Synchronization of heterogeneous oscillator populations in response to weak and strong coupling

Synchronous behavior of a population of chemical oscillators is analyzed in the presence of both weak and strong coupling. In each case, we derive upper bounds on the critical coupling strength which are valid for arbitrary populations of nonlinear, heterogeneous oscillators. For weak perturbations, infinitesimal phase response curves are used to characterize the response to coupling, and graph theoretical techniques are used to predict synchronization. In the strongly perturbed case, we observe a phase dependent perturbation threshold required to elicit an immediate spike and use this behavior for our analytical predictions. Resulting upper bounds on the critical coupling strength agree well with our experimental observations and numerical simulations. Furthermore, important system parameters which determine synchronization are different in the weak and strong coupling regimes. Our results point to new strategies by which limit cycle oscillators can be studied when the applied perturbations become strong enough to immediately reset the phase.

Configurational stability for the Kuramoto-Sakaguchi model

The Kuramoto-Sakaguchi model is a generalization of the well-known Kuramoto model that adds a phase-lag paramater or "frustration" to a network of phase-coupled oscillators. The Kuramoto model is a flow of gradient type, but adding a phase-lag breaks the gradient structure, significantly complicating the analysis of the model. We present several results determining the stability of phase-locked configurations: the first of these gives a sufficient condition for stability, and the second a sufficient condition for instability. In fact, the instability criterion gives a count, modulo 2, of the dimension of the unstable manifold to a fixed point and having an odd count is a sufficient condition for instability of the fixed point. We also present numerical results for both small ( N ≤ 10 ) and large ( N = 50 ) collections of Kuramoto-Sakaguchi oscillators.

Macroscopic models for networks of coupled biological oscillators

The study of synchronization of coupled biological oscillators is fundamental to many areas of biology including neuroscience, cardiac dynamics, and circadian rhythms. Mathematical models of these systems may involve hundreds of variables in thousands of individual cells resulting in an extremely high-dimensional description of the system. This often contrasts with the low-dimensional dynamics exhibited on the collective or macroscopic scale for these systems. We introduce a macroscopic reduction for networks of coupled oscillators motivated by an elegant structure we find in experimental measurements of circadian protein expression and several mathematical models for coupled biological oscillators. The observed structure in the collective amplitude of the oscillator population differs from the well-known Ott-Antonsen ansatz, but its emergence can be characterized through a simple argument depending only on general phase-locking behavior in coupled oscillator systems. We further demonstrate its emergence in networks of noisy heterogeneous oscillators with complex network connectivity. Applying this structure, we derive low-dimensional macroscopic models for oscillator population activity. This approach allows for the incorporation of cellular-level experimental data into the macroscopic model whose parameters and variables can then be directly associated with tissue- or organism-level properties, thereby elucidating the core properties driving the collective behavior of the system.

Modeling and analysis of the impacts of jet lag on circadian rhythm and its role in tumor growth

Circadian rhythms maintain a 24 h oscillation pattern in metabolic, physiological and behavioral processes in all living organisms. Circadian rhythms are organized as biochemical networks located in hypothalamus and peripheral tissues. Rhythmicity in the expression of circadian clock genes plays a vital role in regulating the process of cell division and DNA damage control. The oncogenic protein, MYC and the tumor suppressor, p53 are directly influenced by the circadian clock. Jet lag and altered sleep/wake schedules prominently affect the expression of molecular clock genes. This study is focused on developing a Petri net model to analyze the impacts of long term jet lag on the circadian clock and its probable role in tumor progression. The results depict that jet lag disrupts the normal rhythmic behavior and expression of the circadian clock proteins. This disruption leads to persistent expression of MYC and suppressed expression of p53. Thus, it is inferred that jet lag altered circadian clock negatively affects the expressions of cell cycle regulatory genes and contribute in uncontrolled proliferation of tumor cells.

Neuronal oscillations on an ultra-slow timescale: daily rhythms in electrical activity and gene expression in the mammalian master circadian clockwork

Neuronal oscillations of the brain, such as those observed in the cortices and hippocampi of behaving animals and humans, span across wide frequency bands, from slow delta waves (0.1 Hz) to ultra-fast ripples (600 Hz). Here, we focus on ultra-slow neuronal oscillators in the hypothalamic suprachiasmatic nuclei (SCN), the master daily clock that operates on interlocking transcription-translation feedback loops to produce circadian rhythms in clock gene expression with a period of near 24 h (< 0.001 Hz). This intracellular molecular clock interacts with the cell's membrane through poorly understood mechanisms to drive the daily pattern in the electrical excitability of SCN neurons, exhibiting an up-state during the day and a down-state at night. In turn, the membrane activity feeds back to regulate the oscillatory activity of clock gene programs. In this review, we emphasise the circadian processes that drive daily electrical oscillations in SCN neurons, and highlight how mathematical modelling contributes to our increasing understanding of circadian rhythm generation, synchronisation and communication within this hypothalamic region and across other brain circuits.

Reentrainment of the circadian pacemaker during jet lag: East-west asymmetry and the effects of north-south travel

The normal alignment of circadian rhythms with the 24-h light-dark cycle is disrupted after rapid travel between home and destination time zones, leading to sleep problems, indigestion, and other symptoms collectively known as jet lag. Using mathematical and computational analysis, we study the process of reentrainment to the light-dark cycle of the destination time zone in a model of the human circadian pacemaker. We calculate the reentrainment time for travel between any two points on the globe at any time of the day and year. We construct one-dimensional entrainment maps to explain several properties of jet lag, such as why most people experience worse jet lag after traveling east than west. We show that this east-west asymmetry depends on the endogenous period of the traveler's circadian clock as well as daylength. Thus the critical factor is not simply whether the endogenous period is greater than or less than 24 h as is commonly assumed. We show that the unstable fixed point of an entrainment map determines whether a traveler reentrains through phase advances or phase delays, providing an understanding of the threshold that separates orthodromic and antidromic modes of reentrainment. Contrary to the conventional wisdom that jet lag only occurs after east-west travel across multiple time zones, we predict that the change in daylength encountered during north-south travel can cause jet lag even when no time zones are crossed. Our techniques could be used to provide advice to travelers on how to minimize jet lag on trips involving multiple destinations and a combination of transmeridian and translatitudinal travel.

Accelerating recovery from jet lag: prediction from a multi-oscillator model and its experimental confirmation in model animals

The endogenous circadian clock drives oscillations that are completely synchronized with the environmental day–night rhythms with a period of approximately 24 hours. Temporal misalignment between one’s internal circadian clock and the external solar time often occurs in shift workers and long-distance travelers; such misalignments are accompanied by sleep disturbances and gastrointestinal distress. Repeated exposure to jet lag and rotating shift work increases the risk of lifestyle-related diseases, such as cardiovascular complaints and metabolic insufficiencies. However, the mechanism behind the disruption of one’s internal clock is not well understood. In this paper, we therefore present a new theoretical concept called “jet lag separatrix” to understand circadian clock disruption and slow recovery from jet lag based on the mathematical model describing the hierarchical structure of the circadian clock. To demonstrate the utility of our theoretical study, we applied it to predict that re-entrainment via a two-step jet lag in which a four-hour shift of the light-dark cycle is given in the span of two successive days requires fewer days than when given as a single eight-hour shift. We experimentally verified the feasibility of our theory in C57BL/6 strain mice, with results indicating that this pre-exposure of jet lag is indeed beneficial.

Modeling the network dynamics of pulse-coupled neurons

We derive a mean-field approximation for the macroscopic dynamics of large networks of pulse-coupled theta neurons in order to study the effects of different network degree distributions and degree correlations (assortativity). Using the ansatz of Ott and Antonsen [Chaos 18, 037113 (2008)], we obtain a reduced system of ordinary differential equations describing the mean-field dynamics, with significantly lower dimensionality compared with the complete set of dynamical equations for the system. We find that, for sufficiently large networks and degrees, the dynamical behavior of the reduced system agrees well with that of the full network. This dimensional reduction allows for an efficient characterization of system phase transitions and attractors. For networks with tightly peaked degree distributions, the macroscopic behavior closely resembles that of fully connected networks previously studied by others. In contrast, networks with highly skewed degree distributions exhibit different macroscopic dynamics due to the emergence of degree dependent behavior of different oscillators. For nonassortative networks (i.e., networks without degree correlations), we observe the presence of a synchronously firing phase that can be suppressed by the presence of either assortativity or disassortativity in the network. We show that the results derived here can be used to analyze the effects of network topology on macroscopic behavior in neuronal networks in a computationally efficient fashion.

Ott-Antonsen attractiveness for parameter-dependent oscillatory systems

The Ott-Antonsen (OA) ansatz [Ott and Antonsen, Chaos 18, 037113 (2008); Chaos 19, 023117 (2009)] has been widely used to describe large systems of coupled phase oscillators. If the coupling is sinusoidal and if the phase dynamics does not depend on the specific oscillator, then the macroscopic behavior of the systems can be fully described by a low-dimensional dynamics. Does the corresponding manifold remain attractive when introducing an intrinsic dependence between an oscillator's phase and its dynamics by additional, oscillator specific parameters? To answer this, we extended the OA ansatz and proved that parameter-dependent oscillatory systems converge to the OA manifold given certain conditions. Our proof confirms recent numerical findings that already hinted at this convergence. Furthermore, we offer a thorough mathematical underpinning for networks of so-called theta neurons, where the OA ansatz has just been applied. In a final step, we extend our proof by allowing for time-dependent and multi-dimensional parameters as well as for network topologies other than global coupling. This renders the OA ansatz an excellent starting point for the analysis of a broad class of realistic settings.

Introduction to focus issue: Patterns of network synchronization

The study of synchronization of coupled systems is currently undergoing a major surge fueled by recent discoveries of new forms of collective dynamics and the development of techniques to characterize a myriad of new patterns of network synchronization. This includes chimera states, phenomena determined by symmetry, remote synchronization, and asymmetry-induced synchronization. This Focus Issue presents a selection of contributions at the forefront of these developments, to which this introduction is intended to offer an up-to-date foundation.