Impact of dispersed coupling strength on the free running periods of circadian rhythms

Nonmonotonic enhancement of diversity-induced resonance in systems of mobile oscillators

Diversity is omnipresent in natural and synthetic extended systems, the phenomenon of diversity-induced resonance (DIR), wherein a moderate degree of the diversity can provoke an optimal collective response, provides researchers a brand-new strategy to amplify and utilize the weak signal. As yet the relevant advances focus mostly on the ideal situations where the interactions among elements are uncorrelated with the physical proximity of agents. Such a consideration overlooks interactions mediated by the motion of agents in space. Here, we investigate the signal response of an ensemble of spatial mobile heterogeneous bistable oscillators with two canonical interacting modes: dynamic and preset. The oscillators are considered as mass points and perform random walks in a two-dimensional square plane. Under the dynamic scheme, the oscillators can only interact with other oscillators within a fixed vision radius. For the preset circumstance, the interaction among oscillators occurs only when all of them are in a predefined region at the same moment. We find that the DIR can be obtained in both situations. Additionally, the strength of resonance nonmonotonically rises with respect to the increase of moving speed, and the optimal resonance is acquired by an intermediate magnitude of speed. Finally, we propose reduced equations to guarantee the occurrence of such mobility-optimized DIR on the basis of the fast switching approximation theory and also examine the robustness of such phenomenon through the excitable FitzHugh-Nagumo model and a different spatial motion mechanism. Our results reveal for the first time that the DIR can be optimized by the spatial mobility and thus has promising potential application in the communication of mobile agents.

Double resonance induced by group coupling with quenched disorder

Results show that the astrocytes can not only listen to the talk of large assemble of neurons but also give advice to the conversations and are significant sources of heterogeneous couplings as well. In the present work, we focus on such regulation character of astrocytes and explore the role of heterogeneous couplings among interacted neuron-astrocyte components in a signal response. We consider reduced dynamics in which the listening and advising processes of astrocytes are mapped into the form of group coupling, where the couplings are normally distributed. In both globally coupled overdamped bistable oscillators and an excitable FitzHugh-Nagumo (FHN) neuron model, we numerically and analytically demonstrate that two types of bell-shaped collective response curves can be obtained as the ensemble coupling strength or the heterogeneity of group coupling rise, respectively, which can be seen as a new type of double resonance. Furthermore, through the bifurcation analysis, we verify that these resonant signal responses stem from the competition between dispersion and aggregation induced by heterogeneous group and positive pairwise couplings, respectively. Our results contribute to a better understanding of the signal propagation in coupled systems with quenched disorder.

Motif structure for the four subgroups within the suprachiasmatic nuclei affects its entrainment ability

Circadian rhythms of physiological and behavioral activities are regulated by a central clock. This clock is located in the bilaterally symmetrical suprachiasmatic nucleus (SCN) of mammals. Each nucleus contains a light-sensitive group of neurons, named the ventrolateral (VL) part, with the rest of the neurons being insensitive to light, named the dorsomedial (DM) group. While the coupling between the VL and DM subgroups have been investigated quite well, the communication among the four subgroups across the nuclei did not get a lot of attention. In this article, we theoretically analyzed seven motiflike connection patterns to investigate the network of the two nuclei of the SCN as a whole in relation to the function of the SCN. We investigated the entrainment ability of the SCN and found that the entrainment range is larger in the motifs containing a link between the two VL parts across the nuclei, but it is smaller in the motifs that contain a link between the two DM parts across the nuclei. The SCN may strengthen or weaken connections between the left and right nucleus to accomodate changes in external conditions, such as resynchronization after a jet lag, adjustment to photoperiod or for the aging SCN.

Distinct Components of Photoperiodic Light Are Differentially Encoded by the Mammalian Circadian Clock

Seasonal light cycles influence multiple physiological functions and are mediated through photoperiodic encoding by the circadian system. Despite our knowledge of the strong connection between seasonal light input and downstream circadian changes, less is known about the specific components of seasonal light cycles that are encoded and induce persistent changes in the circadian system. Using combinations of 3 T cycles (23, 24, 26 h) and 2 photoperiods per T cycle (long and short, with duty cycles scaled to each T cycle), we investigate the after-effects of entrainment to these 6 light cycles. We measure locomotor behavior duration (α), period (τ), and entrained phase angle (ψ) in vivo and SCN phase distribution (σ_φ), τ, and ψ ex vivo to refine our understanding of critical light components for influencing particular circadian properties. We find that both photoperiod and T-cycle length drive determination of in vivo ψ but differentially influence after-effects in α and τ, with photoperiod driving changes in α and photoperiod length and T-cycle length combining to influence τ. Using skeleton photoperiods, we demonstrate that in vivo ψ is determined by both parametric and nonparametric components, while changes in α are driven nonparametrically. Within the ex vivo SCN, we find that ψ and σ_φ of the PER2∷LUCIFERASE rhythm follow closely with their likely behavioral counterparts (ψ and α of the locomotor activity rhythm) while also confirming previous reports of τ after-effects of gene expression rhythms showing negative correlations with behavioral τ after-effects in response to T cycles. We demonstrate that within-SCN σ_φ changes, thought to underlie α changes in vivo, are induced primarily nonparametrically. Taken together, our results demonstrate that distinct components of seasonal light input differentially influence ψ, α, and τ and suggest the possibility of separate mechanisms driving the persistent changes in circadian behaviors mediated by seasonal light.

Rev-Erbα and Photoreceptor Outer Segments modulate the Circadian Clock in Retinal Pigment Epithelial Cells

Retinal photoreceptor outer segments (POS) are renewed daily through phagocytosis by the adjacent retinal pigment epithelial (RPE) monolayer. Phagocytosis is mainly driven by the RPE circadian clock but the underlying molecular mechanisms remain elusive. Using ARPE-19 (human RPE cell-line) dispersed and monolayer cell cultures, we investigated the influence of cellular organization on the RPE clock and phagocytosis genes. PCR analysis revealed rhythmic expression of clock and phagocytosis genes in all ARPE-19 cultures. Monolayers had a tendency for higher amplitudes of clock gene oscillations. In all conditions ARNTL, CRY1, PER1-2, REV-ERBα, ITGB5, LAMP1 and PROS1 were rhythmically expressed with REV-ERBα being among the clock genes whose expression showed most robust rhythms in ARPE-19 cells. Using RPE-choroid explant preparations of the mPer2^Luc knock-in mice we found that Rev-Erbα deficiency induced significantly longer periods and earlier phases of PER2-bioluminescence oscillations. Furthermore, early phagocytosis factors β₅-Integrin and FAK and the lysosomal marker LAMP1 protein levels are rhythmic. Finally, POS incubation affects clock and clock-controlled phagocytosis gene expression in RPE monolayers in a time-dependent manner suggesting that POS can reset the RPE clock. These results shed some light on the complex interplay between POS, the RPE clock and clock-controlled phagocytosis machinery which is modulated by Rev-Erbα.

Disassortative Network Structure Improves the Synchronization between Neurons in the Suprachiasmatic Nucleus

In mammals, an endogenous clock located in the suprachiasmatic nucleus (SCN) of the brain regulates the circadian rhythms of physiological and behavioral activities. The SCN is composed of about 20,000 neurons that are autonomous oscillators with nonidentical intrinsic periods ranging from 22 h to 28 h. These neurons are coupled through neurotransmitters and synchronized to form a network, which produces a robust circadian rhythm of a uniform period. The neurons, which are the nodes in the network, are known to be heterogeneous in their characteristics, which is reflected in different phenotypes and different functionality. This heterogeneous nature of the nodes of the network leads to the question as to whether the structure of the SCN network is assortative or disassortative. Thus far, the disassortativity of the SCN network has not been assessed and neither have its effects on the collective behaviors of the SCN neurons. In the present study, we build a directed SCN network composed of hundreds of neurons for a single slice using the method of transfer entropy, based on the experimental data. Then, we measured the synchronization degree as well as the disassortativity coefficient of the network structure (calculated by either the out-degrees or the in-degrees of the nodes) and found that the network of the SCN is a disassortative network. Furthermore, a positive relationship is observed between the synchronization degree and disassortativity of the network, which is confirmed by simulations of our modeling. Our finding suggests that the disassortativity of the network structure plays a role in the synchronization between SCN neurons; that is, the synchronization degree increases with the increase of the disassortativity, which implies that a more heterogeneous coupling in the network of the SCN is important for proper function of the SCN.

Heterogeneity in relaxation rate improves the synchronization of oscillatory neurons in a model of the SCN

The circadian rhythms in mammals, that are regulated by the suprachiasmatic nucleus (SCN) of the brain, have been observed even in the absence of a light-dark cycle. The SCN is composed of about 10 000 autonomous neuronal oscillators, which are heterogenous in many oscillatory properties, including the heterogeneity in relaxation rates. Although the relaxation rate affects the entrainability of the SCN as a whole, not much is known about the reasons why the heterogeneity in relaxation rate exists. In the present study, based on a Poincaré model, we examine whether the heterogeneity in the relaxation rate affects the synchronization of the SCN neuronal oscillators under constant darkness. Both our simulations and theoretical results show that the heterogeneity improves the synchronization. Our findings provide an alternative explanation for the existence of the heterogeneity in the SCN neurons and shed light on the effect of neuronal heterogeneity on the collective behavior of the SCN neurons.

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.

Dispersion of the intrinsic neuronal periods affects the relationship of the entrainment range to the coupling strength in the suprachiasmatic nucleus

Living beings on the Earth are subjected to and entrained (synchronized) to the natural 24-h light-dark cycle. Interestingly, they can also be entrained to an external artificial cycle of non-24-h periods. The range of these periods is called the entrainment range and it differs among species. In mammals, the entrainment range is regulated by a main clock located in the suprachiasmatic nucleus (SCN) which is composed of 10 000 neurons in the brain. Previous works have found that the entrainment range depends on the cellular coupling strength in the SCN. In particular, the entrainment range decreases with the increase of the cellular coupling strength, provided that all the neuronal oscillators are identical. However, the SCN neurons differ in the intrinsic periods that follow a normal distribution in a range from 22 to 28 h. In the present study, taking the dispersion of the intrinsic neuronal periods into account, we examined the relationship between the entrainment range and the coupling strength. Results from numerical simulations and theoretical analyses both show that the relationship is altered to be paraboliclike if the intrinsic neuronal periods are nonidentical, and the maximal entrainment range is obtained with a suitable coupling strength. Our results shed light on the role of the cellular coupling in the entrainment ability of the SCN network.

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.

Differences in intrinsic amplitudes of neuronal oscillators improve synchronization in the suprachiasmatic nucleus

In mammals, a main clock located in the suprachiasmatic nucleus (SCN) regulates the ∼24 h rhythms of behavioral and physiological activities exposed to a natural 24 light-dark cycle or even under constant darkness. The rhythms originate from self-sustained oscillations of the SCN neurons, which differ in both intrinsic periods and intrinsic amplitudes. The intrinsic periods and the intrinsic amplitudes were found to be bound to specific regions in the previous experiments. In particular, neurons of smaller amplitudes and larger periods are located in a ventrolateral part, and neurons of larger amplitudes and smaller periods are in a dorsomedial part. In the present study, we examined the effects of the differences in the intrinsic frequencies and the differences in the intrinsic amplitudes of neuronal oscillators on the synchronization, respectively. We found that the differences in the intrinsic frequencies weaken the synchronization, whereas the differences in the intrinsic amplitudes strengthen the synchronization. Our finding may shed light on the effects of the heterogenous properties of individual neurons on the collective behaviors of the SCN network and provide a way to enhance the synchronization.

The asymmetry of the entrainment range induced by the difference in intrinsic frequencies between two subgroups within the suprachiasmatic nucleus

The rhythms of physiological and behavioral activities in mammals, which are regulated by the main clock suprachiasmatic nucleus (SCN) in the brain, can not be only synchronized to the natural 24 h light-dark cycle, but also to cycles with artificial periods. The range of the artificial periods that the animal can be synchronized to is called entrainment range. In the absence of the light-dark cycle, the animal can also maintain the circadian rhythm with an endogenous period close to 24 h. Experiments found that the entrainment range is not symmetrical with respect to the endogenous period. In the present study, an explanation is given for the asymmetry based on a Kuramoto model which describes the neuronal network of the SCN. Our numerical simulations and theoretical analysis show that the asymmetry results from the difference in the intrinsic frequencies between two subgroups of the SCN, as well as the entrainment range is affected by the difference.

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.

Entrainment range of the suprachiasmatic nucleus affected by the difference in the neuronal amplitudes between the light-sensitive and light-insensitive regions

Mammals not only can be synchronized to the natural 24-h light-dark cycle, but also to a cycle with a non-24-h period. The range of the period of the external cycle, for which the animals can be entrained to, is called the entrainment range, which differs among species. The entrainment range as a characteristic of the animal is determined by the main circadian clock, i.e., the suprachiasmatic nucleus (SCN) in the brain. The SCN is composed of ∼10000 heterogeneous neurons, which can be divided into two subgroups, i.e., the ventrolateral subgroup (VL) directly receiving the light information from the retina and relaying the information to the dorsomedial subgroup (DM). Among the SCN neurons, the amplitudes are different; however, it is unclear that the amplitude is related to the location of the neurons in experiments. In the present study, we examined the effect of the difference in the neuronal amplitude between the VL and the DM on the entrainment range of the SCN, based on a mathematical model, i.e., the Poincaré model, which is used to describe the circadian clock. We find that the maximal entrainment range is obtained when the difference is equal to a critical point. If the difference of the amplitudes of the VL neurons to the amplitudes of the DM neurons is smaller than a critical point, with the increase of the difference, the entrainment range of the SCN increases, while if the difference is larger than the critical point, the entrainment range decreases with the increase of the difference. Our finding may give a potential explanation for the diversity of the entrainment range among species.

The effects of non-self-sustained oscillators on the en-trainment ability of the suprachiasmatic nucleus

In mammals, the circadian rhythms of behavioral and physiological activities are regulated by an endogenous clock located in the suprachiasmatic nucleus (SCN). The SCN is composed of ~20,000 neurons, of which some are capable of self-sustained oscillations, while the others do not oscillate in a self-sustainable manner, but show arrhythmic patterns or damped oscillations. Thus far, the effects of these non-self-sustained oscillatory neurons are not fully explored. Here, we examined how the proportion of the non-self-sustained oscillators affects the free running period under constant darkness and the ability to entrain to the light-dark cycle. We find that the proportion does not affect the free running period, but plays a significant role in the range of entrainment. We also find that its effect on the entrainment range depends on the region where the non-self-sustained oscillators are located. If the non-self-sustained oscillatory neurons are situated in the light-sensitive subregion, the entrainment range narrows when the proportion increases. If they are situated in the light-insensitive subregion, however, the entrainment range broadens with the increase of the proportion. We suggest that the heterogeneity within the light-sensitive and light-insensitive subregions of the SCN has important consequences for how the clock works.

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.