Entrainment range of nonidentical circadian oscillators by a light-dark cycle

Light-induced synchronization modulation: Enhanced in weak coupling and attenuated in strong coupling among suprachiasmatic nucleus neurons

Existing experiments demonstrated that constant light has either enhancing or diminishing effects on the behavioral rhythms of mammals, sparking our intense interest in the underlying mechanisms of this paradoxical phenomenon. The influence of constant light on behavioral rhythms involves the regulation of collective neuronal behavior. The robustness of behavioral rhythms stems from the synchronization of neurons. In mammals, the synchronization among neurons is regulated by the suprachiasmatic nucleus (SCN) located in the hypothalamus. Neurons within the SCN exhibit significant heterogeneity. The intrinsic frequency and coupling strength are two fundamental characteristics determining the internal dynamics of the SCN. In this study, the Poincaré model was employed to investigate the impact of constant light on SCN neuronal dynamics. We found that constant light can modulate neuronal synchronization, a phenomenon tightly linked to the critical threshold value of coupling strength among the neurons. Specifically, under weak coupling, constant light enhances neuronal synchronization. Under strong coupling, constant light weakens synchronization among oscillators. Furthermore, higher light intensity results in lengthened periods and reduced amplitudes. Our findings elucidate important underlying mechanisms by which constant light either enhances or diminishes mammalian behavioral rhythms, and provide a new perspective for understanding the complex regulation network of circadian rhythms.

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.

Heterogeneity in GnRH and kisspeptin neurons and their significance in vertebrate reproductive biology

Vertebrate reproduction is essentially controlled by the hypothalamus-pituitary-gonadal (HPG) axis, which is a central dogma of reproductive biology. Two major hypothalamic neuroendocrine cell groups containing gonadotropin-releasing hormone (GnRH) and kisspeptin are crucial for control of the HPG axis in vertebrates. GnRH and kisspeptin neurons exhibit high levels of heterogeneity including their cellular morphology, biochemistry, neurophysiology and functions. However, the molecular foundation underlying heterogeneities in GnRH and kisspeptin neurons remains unknown. More importantly, the biological and physiological significance of their heterogeneity in reproductive biology is poorly understood. In this review, we first describe the recent advances in the neuroendocrine functions of kisspeptin-GnRH pathways. We then view the recent emerging progress in the heterogeneity of GnRH and kisspeptin neurons using morphological and single-cell transcriptomic analyses. Finally, we discuss our views on the significance of functional heterogeneity of reproductive endocrine cells and their potential relevance to reproductive health.

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.

Dependence of the entrainment on the ratio of amplitudes between two subgroups in the suprachiasmatic nucleus

Organisms can be synchronized not only to the natural 24-h light-dark cycle but also to artificial non-24-h cycles. Interestingly, when the period of the cycle is far from 24 h, organisms may show complicated behavioral patterns. For example, exposed to a 22-h light-dark cycle, in behavioral activity of rats, a phenomenon called "dissociation" emerges, i.e., one periodic component shows a 22-h period and the other shows a period close to the endogenous period of the animal (around 24 h). It has been found that these two components are regulated by two subgroups of the suprachiasmatic nucleus (SCN), respectively, with the ventrolateral part regulating the 22-h component and the dorsomedial part regulating the other component. In the present study, based on a mathematical model, we will examine how the ratio of amplitudes between these two subgroups affects the entrainment of the SCN to the external 22-h light-dark cycle. Our results show that the dissociation happens when the ratio is smaller than 1 and the maximal entrainment (synchronization) ability of the SCN to the external cycle is obtained when the ratio is larger than 1. Our finding sheds light on the dissociation between the subgroups and suggests that the heterogeneity in the amplitudes alter the entrainment ability of the SCN.

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.

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.

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.

Entrainment Maps

Circadian oscillators found across a variety of species are subject to periodic external light-dark forcing. Entrainment to light-dark cycles enables the circadian system to align biological functions with appropriate times of day or night. Phase response curves (PRCs) have been used for decades to gain valuable insights into entrainment; however, PRCs may not accurately describe entrainment to photoperiods with substantial amounts of both light and dark due to their reliance on a single limit cycle attractor. We have developed a new tool, called an entrainment map, that overcomes this limitation of PRCs and can assess whether, and at what phase, a circadian oscillator entrains to external forcing with any photoperiod. This is a 1-dimensional map that we construct for 3 different mathematical models of circadian clocks. Using the map, we are able to determine conditions for existence and stability of phase-locked solutions. In addition, we consider the dependence on various parameters such as the photoperiod and intensity of the external light as well as the mismatch in intrinsic oscillator frequency with the light-dark cycle. We show that the entrainment map yields more accurate predictions for phase locking than methods based on the PRC. The map is also ideally suited to calculate the amount of time required to achieve entrainment as a function of initial conditions and the bifurcations of stable and unstable periodic solutions that lead to loss of entrainment.

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.

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

The dominant endogenous clock, named the suprachiasmatic nucleus (SCN), regulates circadian rhythms of behavioral and physiological activity in mammals. One of the main characteristics of the SCN is that the animal maintains a circadian rhythm with a period close to 24 h in the absence of a daily light-dark cycle (called the free running period). The free running period varies among species due to heterogeneity of the SCN network. Previous studies have shown that the heterogeneity in cellular coupling as well as in intrinsic neuronal periods shortens the free running period. Furthermore, as derived from experiments, one neuron's coupling strength is negatively associated with its period. It is unknown what the effects of this association between coupling strength and period are on the free running period and how the heterogeneity in coupling strength influences this free running period. In the present study we found that in the presence of a negative relationship between one neuron's coupling strength and its period, surprisingly, the dispersion of coupling strengths increases the free running period. Our present finding may shed new light on the understanding of the heterogeneous SCN network and provides an alternative explanation for the diversity of free running periods between species.

The proportion of light-responsive neurons determines the limit cycle properties of the suprachiasmatic nucleus

In mammals, the central clock in the suprachiasmatic nucleus (SCN) controls physiological and behavioral circadian rhythms and is entrained to the external light-dark cycle. The ability of the SCN to entrain can be measured by exposing the animal to a light-dark cycle with a duration that deviates from 24 h (T-cycles); a wider entrainment range reflects a higher ability to entrain. The neurons of the SCN are either light responsive or light unresponsive and are mutually synchronized. The coupling and synchronization between individual SCN neurons and between groups of neurons within the SCN influence the SCN's ability to entrain. Some studies suggest that enhanced coupling decreases the entrainment range, whereas others suggest that enhanced coupling increases the entrainment range. The latter results are surprising, as they are not consistent with the prevalent assumption that the SCN is a limit cycle oscillator that has larger phase shifts when the amplitude is smaller. Here, we used the Poincaré and Goodwin models to test entrainment properties using various proportions of neurons that are responsive to an external stimulus. If all neurons receive external input, the SCN shows limit cycle behavior in all conditions. If all neurons do not receive light input, we found that the entrainment range of the SCN was positively related to coupling strength when coupling was weak. When coupling strength was stronger and above a critical value, the entrainment range was negatively correlated with coupling strength. The results obtained from our simulations were confirmed by analytical studies. Thus, the limit cycle behavior of the SCN appears to be critically dependent on the coupling strength among the neurons and the proportion of neurons that respond to the entraining stimulus.