Heterogeneity induces rhythms of weakly coupled circadian neurons

Diversity-induced trivialization and resilience of neural dynamics

Heterogeneity is omnipresent across all living systems. Diversity enriches the dynamical repertoire of these systems but remains challenging to reconcile with their manifest robustness and dynamical persistence over time, a fundamental feature called resilience. To better understand the mechanism underlying resilience in neural circuits, we considered a nonlinear network model, extracting the relationship between excitability heterogeneity and resilience. To measure resilience, we quantified the number of stationary states of this network, and how they are affected by various control parameters. We analyzed both analytically and numerically gradient and non-gradient systems modeled as non-linear sparse neural networks evolving over long time scales. Our analysis shows that neuronal heterogeneity quenches the number of stationary states while decreasing the susceptibility to bifurcations: a phenomenon known as trivialization. Heterogeneity was found to implement a homeostatic control mechanism enhancing network resilience to changes in network size and connection probability by quenching the system's dynamic volatility.

Influence of asymmetric parameters in higher-order coupling with bimodal frequency distribution

We investigate the phase diagram of the Sakaguchi-Kuramoto model with a higher-order interaction along with the traditional pairwise interaction. We also introduce asymmetry parameters in both the interaction terms and investigate the collective dynamics and their transitions in the phase diagrams under both unimodal and bimodal frequency distributions. We deduce the evolution equations for the macroscopic order parameters and eventually derive pitchfork and Hopf bifurcation curves. Transition from the incoherent state to standing wave pattern is observed in the presence of the unimodal frequency distribution. In contrast, a rich variety of dynamical states such as the incoherent state, partially synchronized state-I, partially synchronized state-II, and standing wave patterns and transitions among them are observed in the phase diagram via various bifurcation scenarios, including saddle-node and homoclinic bifurcations, in the presence of bimodal frequency distribution. Higher-order coupling enhances the spread of the bistable regions in the phase diagrams and also leads to the manifestation of bistability between incoherent and partially synchronized states even with unimodal frequency distribution, which is otherwise not observed with the pairwise coupling. Further, the asymmetry parameters facilitate the onset of several bistable and multistable regions in the phase diagrams. Very large values of the asymmetry parameters allow the phase diagrams to admit only the monostable dynamical states.

Cut the noise or couple up: Coordinating circadian and synthetic clocks

Circadian clocks are important to much of life on Earth and are of inherent interest to humanity, implicated in fields ranging from agriculture and ecology to developmental biology and medicine. New techniques show that it is not simply the presence of clocks, but coordination between them that is critical for complex physiological processes across the kingdoms of life. Recent years have also seen impressive advances in synthetic biology to the point where parallels can be drawn between synthetic biological and circadian oscillators. This review will emphasize theoretical and experimental studies that have revealed a fascinating dichotomy of coupling and heterogeneity among circadian clocks. We will also consolidate the fields of chronobiology and synthetic biology, discussing key design principles of their respective oscillators.

The Sakaguchi-Kuramoto model in presence of asymmetric interactions that break phase-shift symmetry

The celebrated Kuramoto model provides an analytically tractable framework to study spontaneous collective synchronization and comprises globally coupled limit-cycle oscillators interacting symmetrically with one another. The Sakaguchi-Kuramoto model is a generalization of the basic model that considers the presence of a phase lag parameter in the interaction, thereby making it asymmetric between oscillator pairs. Here, we consider a further generalization by adding an interaction that breaks the phase-shift symmetry of the model. The highlight of our study is the unveiling of a very rich bifurcation diagram comprising of both oscillatory and non-oscillatory synchronized states as well as an incoherent state: There are regions of two-state as well as an interesting and hitherto unexplored three-state coexistence arising from asymmetric interactions in our model.

Heritable gene expression variability and stochasticity govern clonal heterogeneity in circadian period

A ubiquitous feature of the circadian clock across life forms is its organization as a network of cellular oscillators, with individual cellular oscillators within the network often exhibiting considerable heterogeneity in their intrinsic periods. The interaction of coupling and heterogeneity in circadian clock networks is hypothesized to influence clock’s entrainability, but our knowledge of mechanisms governing period heterogeneity within circadian clock networks remains largely elusive. In this study, we aimed to explore the principles that underlie intercellular period variation in circadian clock networks (clonal period heterogeneity). To this end, we employed a laboratory selection approach and derived a panel of 25 clonal cell populations exhibiting circadian periods ranging from 22 to 28 h. We report that a single parent clone can produce progeny clones with a wide distribution of circadian periods, and this heterogeneity, in addition to being stochastically driven, has a heritable component. By quantifying the expression of 20 circadian clock and clock-associated genes across our clone panel, we found that inheritance of expression patterns in at least three clock genes might govern clonal period heterogeneity in circadian clock networks. Furthermore, we provide evidence suggesting that heritable epigenetic variation in gene expression regulation might underlie period heterogeneity.

How do genetically identical cells exhibit a different circadian phenotype? This study reveals that a single parent clone can produce progeny with a wide distribution of circadian periods and that this heterogeneity, in addition to being stochastically driven, has a heritable component, likely via heritable epigenetic variation in gene expression regulation.

Noise-driven cellular heterogeneity in circadian periodicity

Our findings have revealed a previously unrecognized link between circadian oscillations and intercellular variation and provide experimental evidence that stochastic transcriptional noise contributes significantly to cell-autonomous circadian periodicity. Interestingly, in separate studies, aging and cancer have been associated with increased transcriptional noise and less robust circadian rhythms. Here, we establish a direct association between transcriptional noise and circadian period. These findings may provide additional directions for researchers in the aging and cancer fields. Furthermore, circadian period may also be used as an indicator of variance in heterogeneity research and drug screening for noise control.

Nongenetic cellular heterogeneity is associated with aging and disease. However, the origins of cell-to-cell variability are complex and the individual contributions of different factors to total phenotypic variance are still unclear. Here, we took advantage of clear phenotypic heterogeneity of circadian oscillations in clonal cell populations to investigate the underlying mechanisms of cell-to-cell variability. Using a fully automated tracking and analysis pipeline, we examined circadian period length in thousands of single cells and hundreds of clonal cell lines and found that longer circadian period is associated with increased intercellular heterogeneity. Based on our experimental results, we then estimated the contributions of heritable and nonheritable factors to this variation in circadian period length using a variance partitioning model. We found that nonheritable noise predominantly drives intercellular circadian period variation in clonal cell lines, thereby revealing a previously unrecognized link between circadian oscillations and intercellular heterogeneity. Moreover, administration of a noise-enhancing drug reversibly increased both period length and variance. These findings suggest that circadian period may be used as an indicator of cellular noise and drug screening for noise control.

Globally fixed-time synchronization of coupled neutral-type neural network with mixed time-varying delays

This paper mainly studies the globally fixed-time synchronization of a class of coupled neutral-type neural networks with mixed time-varying delays via discontinuous feedback controllers. Compared with the traditional neutral-type neural network model, the model in this paper is more general. A class of general discontinuous feedback controllers are designed. With the help of the definition of fixed-time synchronization, the upper right-hand derivative and a defined simple Lyapunov function, some easily verifiable and extensible synchronization criteria are derived to guarantee the fixed-time synchronization between the drive and response systems. Finally, two numerical simulations are given to verify the correctness of the results.

Measuring Relative Coupling Strength in Circadian Systems

Modern imaging techniques allow the monitoring of circadian rhythms of single cells. Coupling between these single cellular circadian oscillators can generate coherent periodic signals on the tissue level that subsequently orchestrate physiological outputs. The strength of coupling in such systems of oscillators is often unclear. In particular, effects on coupling strength by varying cell densities, by knockouts, and by inhibitor applications are debated. In this study, we suggest to quantify the relative coupling strength via analyzing period, phase, and amplitude distributions in ensembles of individual circadian oscillators. Simulations of different oscillator networks show that period and phase distributions become narrower with increasing coupling strength. Moreover, amplitudes can increase due to resonance effects. Variances of periods and phases decay monotonically with coupling strength, and can serve therefore as measures of relative coupling strength. Our theoretical predictions are confirmed by studying recently published experimental data from PERIOD2 expression in slices of the suprachiasmatic nucleus during and after the application of tetrodotoxin (TTX). On analyzing the corresponding period, phase, and amplitude distributions, we can show that treatment with TTX can be associated with a reduced coupling strength in the system of coupled oscillators. Analysis of an oscillator network derived directly from the data confirms our conclusions. We suggest that our approach is also applicable to quantify coupling in fibroblast cultures and hepatocyte networks, and for social synchronization of circadian rhythmicity in rodents, flies, and bees.

Plasticity in the Period of the Circadian Pacemaker Induced by Phase Dispersion of Its Constituent Cellular Clocks

The mammalian circadian pacemaker is commonly thought to be a rigid oscillator that generates output under a variety of circumstances that differ only in phase, period, and/or amplitude. Yet the pacemaker is composed of many cells that each can respond to varying circumstances in different ways. Computer simulations demonstrate that networks of such pacemaker cells behave differently under a light-dark cycle compared with constant darkness. The differences demonstrate that the circadian pacemaker is plastic: The pacemaker shapes its properties in response to the circumstances. A consequence is that properties of a pacemaker under a light-dark cycle cannot be derived from studies of the same system in constant darkness. In this paper we show that the dispersion of phase in a network of coupled oscillators can influence ensemble period: For the considered type of coupling, it is demonstrated that the more synchronous the cells are, the longer is the ensemble period. This is consistent with various data sets obtained in mammals, and even with a data set from fruit flies, in which circadian variation in behavior is regulated in a distinctly differently way from that in mammals. We conclude that environmental circumstances such as photoperiod and exposure to light pulses in otherwise darkness modify the phase distribution of the network and, thereby, the period of the ensemble. Our study supports the view that such properties as circadian period are not solely determined by clock genes but are also determined by the genes that regulate the communication in cellular networks.

Dissociation between two subgroups of the suprachiasmatic nucleus affected by the number of damped oscillated neurons

In mammals, the main clock located in the suprachiasmatic nucleus (SCN) of the brain synchronizes the body rhythms to the environmental light-dark cycle. The SCN is composed of about 2×10^{4} neurons which can be classified into three oscillatory phenotypes: self-sustained oscillators, damped oscillators, and arrhythmic neurons. Exposed to an artificial external light-dark cycle with a period of 22h instead of 24h, two subgroups of the SCN can become desynchronized (dissociated). The ventrolateral (VL) subgroup receives photic input and is entrained to the external cycle and a dorsomedial (DM) subgroup oscillates with its endogenous (i.e., free running) period and is synchronized to the external light-dark cycle through coupling from the VL. In the present study, we examined the effects of damped oscillatory neurons on the dissociation between VL and DM under an external 22h cycle. We found that, with increasing numbers of damped oscillatory neurons located in the VL, the dissociation between the VL and DM emerges, but if these neurons are increasingly present in the DM the dissociation disappears. Hence, the damped oscillatory neurons in different subregions of the SCN play distinct roles in the dissociation between the two subregions of the SCN. This shows that synchrony between SCN subregions is affected by the number of damped oscillatory neurons and the location of these cells. We suggest that more knowledge on the number and the location of these cells may explain why some species do show a dissociation between the subregions and others do not, as the distribution of oscillatory types of neurons offers a plausible and novel candidate mechanism to explain heterogeneity.

Synchronization in slowly switching networks of coupled oscillators

Networks whose structure of connections evolves in time constitute a big challenge in the study of synchronization, in particular when the time scales for the evolution of the graph topology are comparable with (or even longer than) those pertinent to the units’ dynamics. We here focus on networks with a slow-switching structure, and show that the necessary conditions for synchronization, i.e. the conditions for which synchronization is locally stable, are determined by the time average of the largest Lyapunov exponents of transverse modes of the switching topologies. Comparison between fast- and slow-switching networks allows elucidating that slow-switching processes prompt synchronization in the cases where the Master Stability Function is concave, whereas fast-switching schemes facilitate synchronization for convex curves. Moreover, the condition of slow-switching enables the introduction of a control strategy for inducing synchronization in networks with arbitrary structure and coupling strength, which is of evident relevance for broad applications in real world systems.

The synchronization of neuronal oscillators determined by the directed network structure of the suprachiasmatic nucleus under different photoperiods

The main function of the principal clock located in the suprachiasmatic nucleus (SCN) of mammals is synchronizing the body rhythms to the 24 h light-dark cycle. Additionally, the SCN is able to adapt to the photoperiod of the cycle which varies among seasons. Under the long photoperiod (LP), the synchronization degree of the SCN neurons is lower than that under the photoperiod (SP). In the present study, a potential explanation is given for this phenomenon. We propose that the asymmetrical coupling between the light-signal-sensitive part (the ventralateral part, abbreviation: VL) and the light-signal-insensitive part (the dorsalmedial part, abbreviation: DM) of the SCN plays a role in the synchronization degree, which is reflected by the ratio of the number of the directed links from the VL neurons to the DM neurons to the total links of both directions between the VL and the DM. The ratio is assumed to characterize the directed network structure under different photoperiods, which is larger under the SP and smaller under the LP. We found that with the larger ratio in the situation of the SP, the synchronization degree is higher. Our finding may shed new light on the asymmetrical coupling between the VL and the DM, and the network structure of the SCN.

The circadian rhythm induced by the heterogeneous network structure of the suprachiasmatic nucleus

In mammals, the master clock is located in the suprachiasmatic nucleus (SCN), which is composed of about 20 000 nonidentical neuronal oscillators expressing different intrinsic periods. These neurons are coupled through neurotransmitters to form a network consisting of two subgroups, i.e., a ventrolateral (VL) subgroup and a dorsomedial (DM) subgroup. The VL contains about 25% SCN neurons that receive photic input from the retina, and the DM comprises the remaining 75% SCN neurons which are coupled to the VL. The synapses from the VL to the DM are evidently denser than that from the DM to the VL, in which the VL dominates the DM. Therefore, the SCN is a heterogeneous network where the neurons of the VL are linked with a large number of SCN neurons. In the present study, we mimicked the SCN network based on Goodwin model considering four types of networks including an all-to-all network, a Newman-Watts (NW) small world network, an Erdös-Rényi (ER) random network, and a Barabási-Albert (BA) scale free network. We found that the circadian rhythm was induced in the BA, ER, and NW networks, while the circadian rhythm was absent in the all-to-all network with weak cellular coupling, where the amplitude of the circadian rhythm is largest in the BA network which is most heterogeneous in the network structure. Our finding provides an alternative explanation for the induction or enhancement of circadian rhythm by the heterogeneity of the network structure.