Advertisement
Browse Subject Areas
?

Click through the PLOS taxonomy to find articles in your field.

For more information about PLOS Subject Areas, click here.

  • Loading metrics

Photic Desynchronization of Two Subgroups of Circadian Oscillators in a Network Model of the Suprachiasmatic Nucleus with Dispersed Coupling Strengths

  • Changgui Gu,

    Affiliations Institute of Theoretical Physics and Department of Physics, East China Normal University, Shanghai, China, Department of Neurology, University of Massachusetts Medical School, Worcester, Massachusetts, United States of America

  • Zonghua Liu ,

    zhliu@phy.ecnu.edu.cn

    Affiliation Institute of Theoretical Physics and Department of Physics, East China Normal University, Shanghai, China

  • William J. Schwartz,

    Affiliation Department of Neurology, University of Massachusetts Medical School, Worcester, Massachusetts, United States of America

  • Premananda Indic

    Affiliation Department of Neurology, University of Massachusetts Medical School, Worcester, Massachusetts, United States of America

Photic Desynchronization of Two Subgroups of Circadian Oscillators in a Network Model of the Suprachiasmatic Nucleus with Dispersed Coupling Strengths

  • Changgui Gu, 
  • Zonghua Liu, 
  • William J. Schwartz, 
  • Premananda Indic
PLOS
x

Abstract

The suprachiasmatic nucleus (SCN) is the master circadian clock in mammals and is composed of thousands of neuronal oscillators expressing different intrinsic periods. These oscillators form a coupled network with a free-running period around 24 h in constant darkness and entrainable to the external light-dark cycle (T cycle). Coupling plays an important role in setting the period of the network and its range of entrainment. Experiments in rats have shown that two subgroups of oscillators within the SCN, a ventrolateral (VL) subgroup that receives photic input and a dorsomedial (DM) subgroup that is coupled to VL, can be desynchronized under a short (22-h) T cycle, with VL entrained to the cycle and DM free-running. We use a modified Goodwin model to understand how entrainment of the subgroups to short (22-h) and long (26-h) T cycles is influenced by light intensity, the proportion of neurons that receives photic input, and coupling heterogeneity. We find that the model’s critical value for the proportion of photically-sensitive neurons is in accord with actual experimental estimates, while the model’s inclusion of dispersed coupling can account for the experimental observation that VL and DM desynchronize more readily under the 22-h than under the 26-h T cycle. Heterogeneous intercellular coupling within the SCN is likely central to the generation of complex behavioral patterns.

Introduction

Circadian (∼24 h) rhythms in physiological and behavioral measures are universal in living things, reflecting the period of the earth’s rotation. In mammals, circadian rhythms are regulated by a master clock in the suprachiasmatic nucleus (SCN) of the hypothalamus, composed of approximately 20,000 neuronal oscillators; SCN neurons are nonidentical, express different intrinsic periods, and are coupled together to form a network with a coherent output [1]. The period of the network’s output signal is adaptable. Under constant darkness, the rhythm has a free-running period close to 24 h; whereas under an external light-dark cycle (T cycle), it is precisely entrained to a period identical to the external cycle.

The SCN network is heterogeneous [2], [3], [4]. It can be divided into distinct functional subgroups, including a ventrolateral part (VL), which receives photic input from the retina, and a dorsomedial part (DM), which is coupled to VL; both VL and DM contribute to the generation of overt circadian rhythms in physiological and behavioral measures. Peptide neurotransmitters differ between the VL and DM subdivisions, with neurons that express vasoactive intestinal polypeptide (VIP) in the VL and arginine vasopressin in the DM. Periods may vary in different regions of the SCN, with DM running faster than VL in tissue slices [5]. Gamma aminobutyric acid (GABA) neurons are present throughout the SCN and may play a role in coupling the two subdivisions [6]. It has been shown that the circadian oscillation between VL and DM can desynchronize with exposure to short T cycles [7] or after a phase shift of the light-dark cycle [6], [8], [9]; the VL appears to set the final phase of the SCN after the phase shift [6], [8], [9].

Much experimental [10], [11], [12] and theoretical [13], [14], [15] work has been motivated by a desire to understand how this heterogeneous SCN network is reliably entrained and able to generate a coherent output signal, and neuropeptidergic mechanisms appear to be necessary elements [16], [17], [18]. Modeling studies suggest that the circadian clock’s free-running period is proportional to the average intercellular coupling strength [13] and that coupling governs the clock’s range of entrainment to T cycles [15]. However, coupling strength between cells in the SCN network is unlikely to be uniform. The effects of heterogeneous coupling on network synchronization have been studied previously in multi-oscillator models. Daido considered the dispersion of coupling strengths in the Kuramoto model and studied the synchronization property of the network [19], [20]; coupling strength between two oscillators was chosen from a normal distribution. Hong and Strogatz considered a heterogeneous network with excitatory (positive) and inhibitory (negative) coupling in the Kuramoto model to understand the relative contributions of excitatory and inhibitory properties on network synchronization [21]. Our recent work (C.G. and Z.L.) has demonstrated that the dispersion of coupling strengths between SCN cellular oscillators can influence the emergent free running period of the network [22]. To our knowledge, however, there has been no work on the relationship between coupling dispersion and network entrainment.

We examine this issue in the present work, inspired by an interesting experiment performed by de la Iglesia et al. [7] in which rats were exposed to an artificially short 22-h T cycle (11 h light alternating with 11 h darkness). Individual animals expressed two separate circadian motor activity rhythms, with one rhythm entrained by the light and oscillating with a period equal to the external cycle, while the other was not entrained and expressed a period greater than 24 h. Analyses of SCN gene expression suggested that these two motor activity rhythms reflected the stable forced desynchronization of VL and DM subdivisions, respectively. Here we model how entrainment of the subdivisions is influenced by coupling dispersion, as well as by the proportion of cellular oscillators that receive photic input (i.e., that are within VL) and the light intensity.

We use the Goodwin model, a network model of coupled oscillators that has been widely used to describe the mammalian circadian clock [13], [22], [23], [24], [25], [26] (defined in Methods). An individual cellular oscillator of the Goodwin model has three variables: a clock gene mRNA, a clock protein, and a transcriptional inhibitor, all of which form a transcription-translation negative feedback loop. It is assumed that light induces the clock gene mRNA, that a neurotransmitter is increased by the clock gene mRNA, and that neurotransmitters from different neurons form a mean field that couples the neurons together. We consider that neurons receive photic input, where is the total number of neurons in the SCN network and is the ratio of the number of VL neurons to the total number of SCN neurons. We take T cycles of 22 h and 26 h as examples, i.e., symmetrically distant from 24 h. We chose mean field coupling for all of the neurons in the Goodwin model. The coupling strength of all the neurons satisfies a normal distribution with mean value and deviation .

Results

T-cycle Entrainment of an SCN Network without Dispersion of Coupling Strengths

To determine the effects of and light intensity, , on the entrainment of VL and DM to T cycles, we have numerically simulated the Goodwin model with no dispersion of coupling strengths, i.e., . Figure 1 shows the mean field time series of VL and DM oscillations in the 22-h light-dark cycle. Similar to previous observations [13], we find that the time series show quasi-periodic behavior with low light intensity. In (A), the behavior of VL follows the 22-h cycle and sustains a stable phase relationship to it, while the behavior of DM loses its phase relationship to the cycle and runs with a period close to the intrinsic period of the network. This dissociation mimics the forced desynchronization of motor activity rhythms in rats under such a T cycle, as noted previously [7], [27]. When is increased, both VL and DM can be entrained, as in (B). Here the peak of the mean field time series of VL appears around the onset of darkness, whereas that of the DM is phase delayed. This change also can be implemented by . If is reduced, neither VL nor DM entrain to the 22-h cycle. If is increased, both VL and DM can be entrained, as in (B). In sum, both the number of neurons receiving light and the light intensity are important factors for entrainment of the entire SCN network to the T-cycle.

thumbnail
Figure 1. Mean field oscillations of VL and DM during a 22-h T cycle.

(A) VL follows the T cycle, whereas DM free runs for the parameters and . (B) Both VL and DM follow the T cycle for the parameters and . The dispersion of the coupling strengths, η, is set to zero in both (A) and (B). The grey bar indicates the dark phase, and the white bar the light phase, of the T cycle.

https://doi.org/10.1371/journal.pone.0036900.g001

To understand the influence of the parameters and on entrainment, we have calculated the phase diagram of the period of the mean fields of VL and DM in the - plane under short and long T cycles, i.e., of 22 h and 26 h, with (Figure 2). (A) and (C) show that the behavior of VL follows the T cycle for all values of , provided that the light intensity is greater than a critical value, such as . When , the period of VL may not be entrained to the T cycle, depending on ; for example, in Fig. 2A, the period of VL can be 23 h or 24 h. For DM to follow the cycle, however, also must be larger than some threshold; that is, there must be a sufficient number of light-receiving neurons in VL in order to drive the neurons in DM. For a given level of , increasing may allow the entire network to entrain to the driving T cycle. Surprisingly, when is decreased under the 26-h T cycle (D), there is a threshold of at which the period of DM suddenly jumps to a value of 20.8 h to form a locking ratio of 4∶5 with the 26-h T cycle; with further decreased , the period monotonically increases to reach a value of 24 h at .

thumbnail
Figure 2. Period of the mean fields of VL and DM in the pKf plane.

The case for the 22-h T cycle is shown for VL (A) and DM (B), and the case for the 26-h T cycle is shown for VL (C) and DM (D). The coupling strengths are identical for all the oscillators (i.e., η = 0). Entrainment of the sub-network to the 22-h cycle is represented by the yellow region, and entrainment of the sub-network to the 26-h cycle is represented by the blue region.

https://doi.org/10.1371/journal.pone.0036900.g002

A comparison of (B) and (D) shows that the threshold of for entrainment of DM to the 26 h T cycle is greater than that for the 22 h cycle, suggesting that desynchronization between VL and DM might be more likely under long than under short T cycles. Experimentally, however, this appears not to be the case [28], [29], prompting us to consider the influence of heterogeneous coupling strengths on the behavior of the SCN network.

T-cycle Entrainment of an SCN Network with Dispersion of Coupling Strengths

Figure 3 shows the phase diagram of the period of the mean fields of VL and DM in the - plane using two values for . Although qualitatively similar to the diagrams in Figure 2, there are quantitative differences when network coupling strengths are dispersed. In the case of the 22-h T cycle, entrainment is only modestly affected; in contrast, in the case of the 26-h T cycle, increased coupling dispersion significantly reduces the critical value of for DM entrainment, suggesting that the network can be entrained to the long T cycle with a relatively lower .

thumbnail
Figure 3. Effect of coupling dispersion on the period of the mean fields of VL and DM in the pKf

plane. The case for the 22-h T cycle is shown for VL (A) and DM (B) with and for VL (C) and DM (D) with . The corresponding case for the 26-h T cycle is represented in (E) - (H).

https://doi.org/10.1371/journal.pone.0036900.g003

For weak and different values of , we find that the critical () for the 22-h and 26-h T cycles reaches approximately the values of 0.28 and 0.20, respectively. Figure 4 represents the variation in for different values of and . In the case of the 22-h T cycle (A), there is little variation in for different , e.g., is between 0.14 and 0.24 for a . However, does change significantly in the case of the 26-h T-cycle (B). Thus, dispersion of coupling strengths affects entrainment in an asymmetric way, with an influence that is larger for the long than for the short T cycle.

thumbnail
Figure 4. Effect of coupling dispersion on the critical p.

Shown are the cases for the 22-h (A) and 26-h (B) T cycles.

https://doi.org/10.1371/journal.pone.0036900.g004

Instead of randomly assigning coupling values to the network, we also studied the network by selectively assigning coupling values. In two separate trials, we assigned the strongest coupling values to either VL or DM. The phase diagram of the period of the mean fields of VL and DM in the plane was similar to that previously reported.

We also simulated the network with dispersed oscillator periods, rather than dispersed coupling strengths, by selecting different values for the standard deviation of period () for each individual oscillator, such that the period distribution has a mean of 24 h with variability. Without dispersed coupling, we do not observe DM entrainment to the long T cycle at any until is increased to a value greater than 5 h. Since such a large non-identical intrinsic period is not realistic, the dispersion of coupling strengths is likely a crucial factor affecting the entrainment of the network to different T cycles.

Importantly, dispersion of coupling influences the mean field amplitude (Figure 5). The amplitude of VL as a function of changes modestly as the dispersion is increased in the 22-h as well as the 26-h T cycle (A and C). On the other hand, for DM in the 22-h T cycle (B), amplitude decreases as increases. In the 26-h T cycle (D), DM amplitude as a function of changes dramatically, with relatively diminished amplitude as is increased; dispersion counteracts this effect. The enhancement of DM amplitude by increased in the 26-h cycle could be due to enhanced phase synchronization of the oscillators in the network, increased amplitude of the individual oscillators, or both. To begin to distinguish among these possibilities, we studied the effect of dispersed coupling on the order parameter, a measure that represents phase synchronization of the network.

thumbnail
Figure 5. Effect of coupling dispersion on the amplitude of the mean fields of VL and DM.

The case for the 22-h T cycle is shown for VL (A) and DM (B), and the case for the 26-h T cycle is shown for VL (C) and DM (D).

https://doi.org/10.1371/journal.pone.0036900.g005

Effect of Coupling Dispersion on the Order Parameter of the Network

Order parameter characterizes the synchronization property of a network [30], [31], and it is defined here by estimating the phases of the oscillators in VL and DM (see Methods). The order parameter will be unity if all oscillators in the network are perfectly synchronized and zero if they are completely uncorrelated. When their behavior is between these two extremes, the order parameter will be in (0, 1), i.e., representing a phase difference between VL and DM or desynchronization of individual oscillators within VL and/or DM.

We have studied the influence of on the order parameter. Figure 6 shows the dependence of order parameter on the parameters and in the plane. To reveal the effect of coupling dispersion, we have considered two cases, one with and the other with . Under the 22-h T cycle, coupling dispersion reduces for larger and values; whereas under the 26-h T cycle, coupling dispersion enhances. As and increase from (0,0), the relationship between VL and DM changes; comparison of Figure 6 with Figures 2 and 3 visualizes the regions where , i.e., either when VL and DM express different periods or when VL and DM express the same period but with a large phase difference between them. Thus, for the 22-h cycle, although higher and values enhance both VL and DM entrainment to the cycle, the reduction of with dispersed coupling suggests that individual oscillators are not fully synchronized within the network, with a greater vulnerability to perturbations of light intensity. For the 26-h cycle, coupling dispersion synchronizes network oscillation for >0.6; the gradual increase of DM mean field amplitude as increases further (Figure 5D) is thus attributable to increased individual oscillator amplitude.

thumbnail
Figure 6. Effect of coupling dispersion on the order parameter of the network.

The case for the 22-h T cycle is shown with coupling dispersion (A) and (B), and the case for the 26-h T cycle is shown with coupling dispersion (C) and (D).

https://doi.org/10.1371/journal.pone.0036900.g006

These considerations imply that VL and DM desynchronize more readily under the 22-h than under the 26-h T cycle and that dispersion of coupling strengths improves network robustness preferentially under the 26-h cycle.

Effect of Coupling Dispersion on the Network’s Phase Response Curve to a Light Pulse

The network’s capacity to generate phase advances or delays can be quantified as a phase-response curve (PRC), measured by plotting the phase shifts that occur in the rhythm when discrete light pulses are applied at different phase points across the circadian cycle [24], [32], [33], [34]. Figure 7 represents the family of PRC’s obtained to a 1-h light pulse of increasing intensities, showing that the phase response region (i.e., the area under the delay and advance zones) increases in magnitude with increasing . Notably, as the value of increases, the area under the delay zone increases relatively more than that under the advance zone, as calculated in Table 1, where represents the ratio of the area under the delay zone to the area under the advance zone. Advances should correspond to the capacity of the network to follow a T cycle less than 24 h, while delays should correspond to its capacity to follow a T cycle greater than 24 h [35].

thumbnail
Figure 7. Effect of coupling dispersion on the phase response curve (PRC) of the network.

Shown are PRCs with coupling dispersions (A), (B), and (C). Although the network shows relatively larger phase advances and delays with increased coupling dispersion, the area under the phase delay zones is greater than that under the advance zones. The PRCs were similar for all the values of .

https://doi.org/10.1371/journal.pone.0036900.g007

thumbnail
Table 1. Effect of coupling dispersion on the ratio of the area of the delay zone to the advance zone.

https://doi.org/10.1371/journal.pone.0036900.t001

Discussion

Here we analyze the photic desynchronization of two subgroups of circadian oscillators in a network model of the suprachiasmatic nucleus. As also demonstrated in experiments with rats exposed to a short T cycle of low light intensity [7], [36], a subgroup of oscillators receiving photic input (VL) can entrain to the external cycle while the other, coupled subgroup (DM) expresses an unentrained period greater than 24 h.

Granada et al. [37] have modeled this forced desynchronization of rat activity rhythms as a single oscillator with oscillatory interactions (modulation and superposition) between the external cycle and the internal clock, while Schwartz et al. [38] have modeled entrainment to the T cycle by two coupled oscillators forced by a Zeitgeber. Casiraghi et al. [39] have used a two oscillator model to analyze a chronic jet lag paradigm that leads to forced desynchrony, and they observed an asymmetry in its behavior similar to our findings reported here. We have taken the Goodwin model and extended it to include , the proportion of all SCN cellular oscillators that receive photic input, and , the dispersion of coupling strengths. We find, first, that network desynchronization (with an entrained VL but an unentrained DM) depends on light intensity and the value of . Relatively higher light intensities protect the network from desynchronization, as reported experimentally [36]. Experiments estimate that the value of for the rodent SCN ranges from 20% to 33%, based on molecular, electrophysiological, and computational studies [40], [41]. Comparing these results to our simulations in Figure 4, we find that is a good parameter value to fit the experiments. At this value, there is a critical value of for network entrainment to short (22 h) and long (26 h) T cycles of 0.28 and 0.20, respectively, and the critical value appears fairly insensitive to . We predict that the rat SCN is likely to have this very large heterogeneity in coupling strengths, given that the critical best matches the experimental estimate for the larger values of .

Second, we find that the inclusion of dispersed coupling strengths affects network entrainment in a preferential manner, such that increased significantly reduces the critical value of for DM entrainment to the 26-h T cycle. A consequence of this influence is that network robustness is superior under the 26-h cycle while desynchronization is favored under the 22-h cycle. In fact, such an asymmetry has been observed experimentally, with no obvious desynchronization of rat motor activity rhythms during exposure to long T cycles [28].

The basis for this asymmetry is unclear. Coupling dispersion appears to generally increase the effect of light on the system, and since the delay zone of the PRC is greater than the advance zone, a preferential effect on entrainment to the 26-h T cycle might be expected. But such an explanation does not account for the u-shaped, rather than monotonic, function for the 22-h T cycle. Moreover, differences in entrainment to short and long T cycles may not be a general feature of the SCN; it may not be true for other species [42], and the possible effects of locomotor activity itself on SCN network behavior (e.g., in or out of a running wheel, diurnal or nocturnal activity pattern) needs further investigation.

Our findings may provide new elements to the theory of coupled oscillators, especially with regard to chimera states in which one group of the system is synchronized and the other is desynchronized [43], [44], [45], [46], [47]; in these studies, the oscillators in the two groups are identical and the chimera states are generally induced by the initial conditions. However, in our case, the discovery that the subgroup VL may be entrained to the T cycle while the group DM remains free-running is similar to the chimera state, but this phenomenon does not depend on the initial conditions. Thus, a novel oscillator theory is needed to explain this robustness to initial conditions and should be a topic for further studies.

Heterogeneous intercellular coupling within the SCN is likely central to the generation of complex behavioral patterns. Non-uniform SCN network architecture also has been implicated in the phase-“splitting” of locomotor activity cycles seen in hamsters maintained in constant environmental light [48]. In the future, we hope to consider how topology influences entrainment, in contrast to the mean field used here.

Methods

We represent each mammalian cell of the network as a Goodwin oscillator. The Goodwin model is a widely used mathematical model to represent the behavior of the gene regulatory network in single cellular circadian oscillators [13]. The model represents the transcription-translation behavior of the single cell by using three variables that include a clock gene mRNA (), a clock protein (), and a transcriptional inhibitor ().

As our network model, we consider the mean field-modified Goodwin model proposed by [13] with a global coupling strength. The modified Goodwin model with oscillators is represented as follows:(1)Where the state variables , , represent the concentrations, respectively, of a clock gene mRNA, a clock protein and a transcriptional inhibitor in each clock cell . Neurotransmitter is induced by the mRNA . The intercellular coupling is implemented through the neurotransmitter which acts as a mean field of , the coupling strength represents the sensitivity of the individual oscillator to the neurotransmitter and is required to be a positive value here, and is the light term. We considered the parameters as in [13]:

The coupling strength is different for different oscillators and assumes a value from a normal distribution with a mean 0.5 and a standard deviation . When , the network is heterogeneous with distribution of coupling.

In order to understand the dissociation behavior observed under a T cycle outside the range of entrainment, we modified the Goodwin model to include the fact that light acts directly on only a portion of the neurons in the network. Furthermore, the light term that is applied to a fraction neurons with being less than one and positive, is considered to be located in the VL subdivision. Mathematically, the effect of light is represented as:Where TL is the period of the light-dark cycle and Kf is the light intensity.

As pointed out in our previous paper [22], the dispersion of coupling strengths influences the free-running period of the SCN. In order to compare the influence of different coupling dispersions on entrainment of the SCN network to T cycles, it is necessary to make the free-running period the same for different dispersions. To set the free-running period to 24 h, we multiply a rescaling factor to the left hand of equation (1) except for the light term and coupling term, i.e., multiply the same to the parameters for the deviation . For example, is equal to 1.26 for , 1.22 for , 1.16 for , and 1.13 for .

For simplicity, we refer to the network that is comprised of pN neurons as VL and the network comprised of the remaining neurons as DM. To understand the behavior of the VL and DM subdivisions, we define the mean field of VL and DM respectively as

In addition, to understand the synchronization properties between VL and DM, we have estimated the phase of the individual neurons by using the Hilbert transform [49], [50]. From the estimated phase of VL and DM, we introduce an order parameter as:where is the estimated phase from the mean field output time series of VL or DM and denotes average over time. The average of is defined as the angular frequency and the period is obtained by . To determine entrainment of VL or DM to the , we estimated the period of VL or DM and estimated its absolute difference from . If the absolute difference in period is less than 0.25 h, the corresponding subgroup (VL or DM) is considered to be entrained. To numerically calculate the equations, we use the fourth order Runga-Kutta method with time step of 0.1 h. Initial 20000 time steps are neglected to avoid the effect of transients. The number of oscillators is, and the simulations are performed five times, with initial conditions selected randomly from a uniform distribution in the range (0–1) for , , and . We have also calculated the case of and time step of 0.01 h. Two additional simulations are performed with selective assignment of coupling in which larger values are assigned to either VL or DM without changing the intrinsic distribution of .

To obtain the phase-response curve, we applied 1-h light pulses at different phases to the model, with intensity and with different values for . The corresponding advance or delay is detected from the output of the network. Advance corresponds to the capacity of the SCN network to follow a light-dark cycle with period less than the free running period, and delay is the capacity of the network to follow a light-dark cycle with period greater than the free running period [35].

Author Contributions

Conceived and designed the experiments: CG WJS PI. Performed the experiments: CG ZL. Analyzed the data: CG ZL WJS PI. Contributed reagents/materials/analysis tools: CG ZL WJS PI. Wrote the paper: CG ZL WJS PI.

References

  1. 1. Welsh DK, Takahashi JS, Kay SA (2010) Suprachiasmatic nucleus: cell autonomy and network properties. Annu Rev Physiol 72: 551–577.DK WelshJS TakahashiSA Kay2010Suprachiasmatic nucleus: cell autonomy and network properties.Annu Rev Physiol72551577
  2. 2. Silver R, Schwartz WJ (2005) The suprachiasmatic nucleus is a functionally heterogeneous timekeeping organ. Methods Enzymol 393: 451–465.R. SilverWJ Schwartz2005The suprachiasmatic nucleus is a functionally heterogeneous timekeeping organ.Methods Enzymol393451465
  3. 3. Morin LP (2007) SCN organization reconsidered. J Biol Rhythms 22: 3–13.LP Morin2007SCN organization reconsidered.J Biol Rhythms22313
  4. 4. Noguchi T, Watanabe K (2008) Regional differences in circadian period within the suprachiasmatic nucleus. Brain Res 1239: 119–126.T. NoguchiK. Watanabe2008Regional differences in circadian period within the suprachiasmatic nucleus.Brain Res1239119126
  5. 5. Noguchi T, Watanabe K, Ogura A, Yamaoka S (2004) The clock in the dorsal suprachiasmatic nucleus runs faster than that in the ventral. Eur J Neurosci 20: 3199–3202.T. NoguchiK. WatanabeA. OguraS. Yamaoka2004The clock in the dorsal suprachiasmatic nucleus runs faster than that in the ventral.Eur J Neurosci2031993202
  6. 6. Albus H, Vansteensel MJ, Michel S, Block GD, Meijer JH (2005) A GABAergic mechanism is necessary for coupling dissociable ventral and dorsal regional oscillators within the circadian clock. Curr Biol 15: 886–893.H. AlbusMJ VansteenselS. MichelGD BlockJH Meijer2005A GABAergic mechanism is necessary for coupling dissociable ventral and dorsal regional oscillators within the circadian clock.Curr Biol15886893
  7. 7. de la Iglesia HO, Cambras T, Schwartz WJ, Diez-Noguera A (2004) Forced desynchronization of dual circadian oscillators within the rat suprachiasmatic nucleus. Curr Biol 14: 796–800.HO de la IglesiaT. CambrasWJ SchwartzA. Diez-Noguera2004Forced desynchronization of dual circadian oscillators within the rat suprachiasmatic nucleus.Curr Biol14796800
  8. 8. Nagano M, Adachi A, Nakahama K, Nakamura T, Tamada M, et al. (2003) An abrupt shift in the day/night cycle causes desynchrony in the mammalian circadian center. J Neurosci 23: 6141–6151.M. NaganoA. AdachiK. NakahamaT. NakamuraM. Tamada2003An abrupt shift in the day/night cycle causes desynchrony in the mammalian circadian center.J Neurosci2361416151
  9. 9. Nakamura W, Yamazaki S, Takasu NN, Mishima K, Block GD (2005) Differential response of Period 1 expression within the suprachiasmatic nucleus. J Neurosci 25: 5481–5487.W. NakamuraS. YamazakiNN TakasuK. MishimaGD Block2005Differential response of Period 1 expression within the suprachiasmatic nucleus.J Neurosci2554815487
  10. 10. Yamaguchi S, Isejima H, Matsuo T, Okura R, Yagita K, et al. (2003) Synchronization of cellular clocks in the suprachiasmatic nucleus. Science 302: 1408–1412.S. YamaguchiH. IsejimaT. MatsuoR. OkuraK. Yagita2003Synchronization of cellular clocks in the suprachiasmatic nucleus.Science30214081412
  11. 11. Aton SJ, Herzog ED (2005) Come together, right…now: synchronization of rhythms in a mammalian circadian clock. Neuron 48: 531–534.SJ AtonED Herzog2005Come together, right…now: synchronization of rhythms in a mammalian circadian clock.Neuron48531534
  12. 12. Liu AC, Welsh DK, Ko CH, Tran HG, Zhang EE, et al. (2007) Intercellular coupling confers robustness against mutations in the SCN circadian clock network. Cell 129: 605–616.AC LiuDK WelshCH KoHG TranEE Zhang2007Intercellular coupling confers robustness against mutations in the SCN circadian clock network.Cell129605616
  13. 13. Gonze D, Bernard S, Waltermann C, Kramer A, Herzel H (2005) Spontaneous synchronization of coupled circadian oscillators. Biophys J 89: 120–129.D. GonzeS. BernardC. WaltermannA. KramerH. Herzel2005Spontaneous synchronization of coupled circadian oscillators.Biophys J89120129
  14. 14. Bernard S, Gonze D, Cajavec B, Herzel H, Kramer A (2007) Synchronization-induced rhythmicity of circadian oscillators in the suprachiasmatic nucleus. PLoS Comput Biol 3: e68.S. BernardD. GonzeB. CajavecH. HerzelA. Kramer2007Synchronization-induced rhythmicity of circadian oscillators in the suprachiasmatic nucleus.PLoS Comput Biol3e68
  15. 15. Abraham U, Granada AE, Westermark PO, Heine M, Kramer A, et al. (2010) Coupling governs entrainment range of circadian clocks. Mol Syst Biol 6: 438.U. AbrahamAE GranadaPO WestermarkM. HeineA. Kramer2010Coupling governs entrainment range of circadian clocks.Mol Syst Biol6438
  16. 16. Aton SJ, Colwell CS, Harmar AJ, Waschek J, Herzog ED (2005) Vasoactive intestinal polypeptide mediates circadian rhythmicity and synchrony in mammalian clock neurons. Nat Neurosci 8: 476–483.SJ AtonCS ColwellAJ HarmarJ. WaschekED Herzog2005Vasoactive intestinal polypeptide mediates circadian rhythmicity and synchrony in mammalian clock neurons.Nat Neurosci8476483
  17. 17. Maywood ES, Reddy AB, Wong GK, O’Neill JS, O’Brien JA, et al. (2006) Synchronization and maintenance of timekeeping in suprachiasmatic circadian clock cells by neuropeptidergic signaling. Curr Biol 16: 599–605.ES MaywoodAB ReddyGK WongJS O’NeillJA O’Brien2006Synchronization and maintenance of timekeeping in suprachiasmatic circadian clock cells by neuropeptidergic signaling.Curr Biol16599605
  18. 18. Li JD, Burton KJ, Zhang C, Hu SB, Zhou QY (2009) Vasopressin receptor V1a regulates circadian rhythms of locomotor activity and expression of clock-controlled genes in the suprachiasmatic nuclei. Am J Physiol Regul Integr Comp Physiol 296: R824–830.JD LiKJ BurtonC. ZhangSB HuQY Zhou2009Vasopressin receptor V1a regulates circadian rhythms of locomotor activity and expression of clock-controlled genes in the suprachiasmatic nuclei.Am J Physiol Regul Integr Comp Physiol296R824830
  19. 19. Daido H (1992) Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions. Phys Rev Lett 68: 1073–1076.H. Daido1992Quasientrainment and slow relaxation in a population of oscillators with random and frustrated interactions.Phys Rev Lett6810731076
  20. 20. Daido H (2000) Algebraic relaxation of an order parameter in randomly coupled limit-cycle oscillators. Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics 61: 2145–2147.H. Daido2000Algebraic relaxation of an order parameter in randomly coupled limit-cycle oscillators.Phys Rev E Stat Phys Plasmas Fluids Relat Interdiscip Topics6121452147
  21. 21. Hong H, Strogatz SH (2011) Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators. Phys Rev Lett 106: 054102.H. HongSH Strogatz2011Kuramoto model of coupled oscillators with positive and negative coupling parameters: an example of conformist and contrarian oscillators.Phys Rev Lett106054102
  22. 22. Gu C, Wang J, Liu Z (2009) Free-running period of neurons in the suprachiasmatic nucleus: Its dependence on the distribution of neuronal coupling strengths. Phys Rev E Stat Nonlin Soft Matter Phys 80: 030904.C. GuJ. WangZ. Liu2009Free-running period of neurons in the suprachiasmatic nucleus: Its dependence on the distribution of neuronal coupling strengths.Phys Rev E Stat Nonlin Soft Matter Phys80030904
  23. 23. Ruoff P, Vinsjevik M, Monnerjahn C, Rensing L (1999) The Goodwin oscillator: on the importance of degradation reactions in the circadian clock. J Biol Rhythms 14: 469–479.P. RuoffM. VinsjevikC. MonnerjahnL. Rensing1999The Goodwin oscillator: on the importance of degradation reactions in the circadian clock.J Biol Rhythms14469479
  24. 24. Locke JC, Westermark PO, Kramer A, Herzel H (2008) Global parameter search reveals design principles of the mammalian circadian clock. BMC Syst Biol 2: 22.JC LockePO WestermarkA. KramerH. Herzel2008Global parameter search reveals design principles of the mammalian circadian clock.BMC Syst Biol222
  25. 25. Ullner E, Buceta J, Diez-Noguera A, Garcia-Ojalvo J (2009) Noise-induced coherence in multicellular circadian clocks. Biophys J 96: 3573–3581.E. UllnerJ. BucetaA. Diez-NogueraJ. Garcia-Ojalvo2009Noise-induced coherence in multicellular circadian clocks.Biophys J9635733581
  26. 26. Gu C, Wang J, Liu Z (2011) Mechanism of phase splitting in two coupled groups of suprachiasmatic-nucleus neurons. Phys Rev E Stat Nonlin Soft Matter Phys 83: 046224.C. GuJ. WangZ. Liu2011Mechanism of phase splitting in two coupled groups of suprachiasmatic-nucleus neurons.Phys Rev E Stat Nonlin Soft Matter Phys83046224
  27. 27. Campuzano A, Vilaplana J, Cambras T, Diez-Noguera A (1998) Dissociation of the rat motor activity rhythm under T cycles shorter than 24 hours. Physiol Behav 63: 171–176.A. CampuzanoJ. VilaplanaT. CambrasA. Diez-Noguera1998Dissociation of the rat motor activity rhythm under T cycles shorter than 24 hours.Physiol Behav63171176
  28. 28. Vilaplana J, Cambras T, Campuzano A, Diez-Noguera A (1997) Simultaneous manifestation of free-running and entrained rhythms in the rat motor activity explained by a multioscillatory system. Chronobiol Int 14: 9–18.J. VilaplanaT. CambrasA. CampuzanoA. Diez-Noguera1997Simultaneous manifestation of free-running and entrained rhythms in the rat motor activity explained by a multioscillatory system.Chronobiol Int14918
  29. 29. Usui S, Takahashi Y, Okazaki T (2000) Range of entrainment of rat circadian rhythms to sinusoidal light-intensity cycles. Am J Physiol Regul Integr Comp Physiol 278: R1148–1156.S. UsuiY. TakahashiT. Okazaki2000Range of entrainment of rat circadian rhythms to sinusoidal light-intensity cycles.Am J Physiol Regul Integr Comp Physiol278R11481156
  30. 30. Liang X, Tang M, Dhamala M, Liu Z (2009) Phase synchronization of inhibitory bursting neurons induced by distributed time delays in chemical coupling. Phys Rev E Stat Nonlin Soft Matter Phys 80: 066202.X. LiangM. TangM. DhamalaZ. Liu2009Phase synchronization of inhibitory bursting neurons induced by distributed time delays in chemical coupling.Phys Rev E Stat Nonlin Soft Matter Phys80066202
  31. 31. Strogatz S (2000) From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators. Physica D 143: 20.S. Strogatz2000From Kuramoto to Crawford: exploring the onset of synchronization in populations of coupled oscillators.Physica D14320
  32. 32. Daan S, Pittendrigh C (1976) A functional analysis of circadian pacemakers in nocturnal rodents. II. The variability of phase response curves. J Comp Physiol A 106: 253–256.S. DaanC. Pittendrigh1976A functional analysis of circadian pacemakers in nocturnal rodents. II. The variability of phase response curves.J Comp Physiol A106253256
  33. 33. Ruoff P, Vinsjevik M, Monnerjahn C, Rensing L (2001) The Goodwin model: simulating the effect of light pulses on the circadian sporulation rhythm of Neurospora crassa. J Theor Biol 209: 29–42.P. RuoffM. VinsjevikC. MonnerjahnL. Rensing2001The Goodwin model: simulating the effect of light pulses on the circadian sporulation rhythm of Neurospora crassa.J Theor Biol2092942
  34. 34. Granada A, Hennig RM, Ronacher B, Kramer A, Herzel H (2009) Phase response curves elucidating the dynamics of coupled oscillators. Methods Enzymol 454: 1–27.A. GranadaRM HennigB. RonacherA. KramerH. Herzel2009Phase response curves elucidating the dynamics of coupled oscillators.Methods Enzymol454127
  35. 35. Pendergast JS, Friday RC, Yamazaki S (2010) Photic entrainment of period mutant mice is predicted from their phase response curves. J Neurosci 30: 12179–12184.JS PendergastRC FridayS. Yamazaki2010Photic entrainment of period mutant mice is predicted from their phase response curves.J Neurosci301217912184
  36. 36. Cambras T, Vilaplana J, Campuzano A, Canal-Corretger MM, Carulla M, et al. (2000) Entrainment of the rat motor activity rhythm: effects of the light-dark cycle and physical exercise. Physiol Behav 70: 227–232.T. CambrasJ. VilaplanaA. CampuzanoMM Canal-CorretgerM. Carulla2000Entrainment of the rat motor activity rhythm: effects of the light-dark cycle and physical exercise.Physiol Behav70227232
  37. 37. Granada AE, Cambras T, Diez-Noguera A, Herzel H (2011) Circadian desynchronization. Interface Focus 1: 153–166.AE GranadaT. CambrasA. Diez-NogueraH. Herzel2011Circadian desynchronization.Interface Focus1153166
  38. 38. Schwartz MD, Wotus C, Liu T, Friesen WO, Borjigin J, et al. (2009) Dissociation of circadian and light inhibition of melatonin release through forced desynchronization in the rat. Proc Natl Acad Sci U S A 106: 17540–17545.MD SchwartzC. WotusT. LiuWO FriesenJ. Borjigin2009Dissociation of circadian and light inhibition of melatonin release through forced desynchronization in the rat.Proc Natl Acad Sci U S A1061754017545
  39. 39. Casiraghi LP, Oda GA, Chiesa JJ, Friesen WO, Golombek DA (2012) Forced desynchronization of activity rhythms in a model of chronic jet lag in mice. J Biol Rhythms 27: 59–69.LP CasiraghiGA OdaJJ ChiesaWO FriesenDA Golombek2012Forced desynchronization of activity rhythms in a model of chronic jet lag in mice.J Biol Rhythms275969
  40. 40. Rohling JH, vanderLeest HT, Michel S, Vansteensel MJ, Meijer JH (2011) Phase resetting of the mammalian circadian clock relies on a rapid shift of a small population of pacemaker neurons. PLoS One 6: e25437.JH RohlingHT vanderLeestS. MichelMJ VansteenselJH Meijer2011Phase resetting of the mammalian circadian clock relies on a rapid shift of a small population of pacemaker neurons.PLoS One6e25437
  41. 41. Meijer JH, Schwartz WJ (2003) In search of the pathways for light-induced pacemaker resetting in the suprachiasmatic nucleus. J Biol Rhythms 18: 235–249.JH MeijerWJ Schwartz2003In search of the pathways for light-induced pacemaker resetting in the suprachiasmatic nucleus.J Biol Rhythms18235249
  42. 42. Vivanco P, Otalora BB, Rol MA, Madrid JA (2010) Dissociation of the circadian system of Octodon degus by T28 and T21 light-dark cycles. Chronobiol Int 27: 1580–1595.P. VivancoBB OtaloraMA RolJA Madrid2010Dissociation of the circadian system of Octodon degus by T28 and T21 light-dark cycles.Chronobiol Int2715801595
  43. 43. Abrams DM, Mirollo R, Strogatz SH, Wiley DA (2008) Solvable model for chimera states of coupled oscillators. Phys Rev Lett 101: 084103.DM AbramsR. MirolloSH StrogatzDA Wiley2008Solvable model for chimera states of coupled oscillators.Phys Rev Lett101084103
  44. 44. Pikovsky A, Rosenblum M (2008) Partially integrable dynamics of hierarchical populations of coupled oscillators. Phys Rev Lett 101: 264103.A. PikovskyM. Rosenblum2008Partially integrable dynamics of hierarchical populations of coupled oscillators.Phys Rev Lett101264103
  45. 45. Ott E, Antonsen TM (2008) Low dimensional behavior of large systems of globally coupled oscillators. Chaos 18: 037113.E. OttTM Antonsen2008Low dimensional behavior of large systems of globally coupled oscillators.Chaos18037113
  46. 46. Lee WS, Ott E, Antonsen TM (2009) Large coupled oscillator systems with heterogeneous interaction delays. Phys Rev Lett 103: 044101.WS LeeE. OttTM Antonsen2009Large coupled oscillator systems with heterogeneous interaction delays.Phys Rev Lett103044101
  47. 47. Martens EA, Laing CR, Strogatz SH (2010) Solvable model of spiral wave chimeras. Phys Rev Lett 104: 044101.EA MartensCR LaingSH Strogatz2010Solvable model of spiral wave chimeras.Phys Rev Lett104044101
  48. 48. Indic P, Schwartz WJ, Paydarfar D (2008) Design principles for phase-splitting behaviour of coupled cellular oscillators: clues from hamsters with ‘split’ circadian rhythms. J R Soc Interface 5: 873–883.P. IndicWJ SchwartzD. Paydarfar2008Design principles for phase-splitting behaviour of coupled cellular oscillators: clues from hamsters with ‘split’ circadian rhythms.J R Soc Interface5873883
  49. 49. Rosenblum MG, Pikovsky AS, Kurths J (1996) Phase synchronization of chaotic oscillators. Phys Rev Lett 76: 1804–1807.MG RosenblumAS PikovskyJ. Kurths1996Phase synchronization of chaotic oscillators.Phys Rev Lett7618041807
  50. 50. Aylett M, Marples G, Jones K (1999) Home blood pressure monitoring: its effect on the management of hypertension in general practice. Br J Gen Pract 49: 725–728.M. AylettG. MarplesK. Jones1999Home blood pressure monitoring: its effect on the management of hypertension in general practice.Br J Gen Pract49725728