Robust synchronization is a critical feature of several systems including the mammalian circadian clock. The master circadian clock in mammals consists of about 20000 ‘sloppy’ neuronal oscillators within the hypothalamus that keep robust time by synchronization driven by inter-neuronal coupling. The complete understanding of this synchronization in the mammalian circadian clock and the mechanisms underlying it remain an open question. Experiments and computational studies have shown that coupling individual oscillators can achieve robust synchrony, despite heterogeneity and different network topologies. But, much less is known regarding the mechanisms and circuits involved in achieving this coupling, due to both system complexity and experimental limitations. Here, we computationally study the coupling mediated by the primary coupling neuropeptide, vasoactive intestinal peptide (VIP) and its canonical receptor, VPAC2R, using the transcriptional elements and generic mode of VIP-VPAC2R signaling. We find that synchrony is only possible if VIP (an inducer of Per expression) is released in-phase with activators of Per expression. Moreover, anti-phasic VIP release suppresses coherent rhythms by moving the network into a desynchronous state. Importantly, experimentally observed rhythms in VPAC2R have little effect on network synchronization, but can improve the amplitude of the SCN network rhythms while narrowing the network entrainment range. We further show that these findings are valid across several computational network models. Thus, we identified a general design principle to achieve robust synchronization: An activating coupling agent, such as VIP, must act in-phase with the activity of core-clock promoters. More generally, the phase of coupling is as critical as the strength of coupling from the viewpoint of synchrony and entrainment.
Organisms evolved an internal biological timekeeper or ‘clock’ to temporally organize and regulate their physiological processes to best cope with a fundamentally periodic natural environment. In mice, the model system in mammals, this master biological clock consists of 20000 neurons within the suprachiasmatic nucleus (SCN) in the hypothalamus. Each neuron sustains oscillations in the expression of ‘clock genes’ and firing that are noisy and variable. Inter-neuronal coupling [1], [2] lends both precision [3] to the clock via synchronization and robustness to the clock against certain perturbations of the individual neuronal oscillators [4]. Coupling between neurons is mediated by synapses, gap junctions, and neuropeptides, such as vasoactive intestinal peptide (VIP) [5]. Synchronization produced by neuropeptide-based coupling is the central focus of this work. VIP is one of the most important and well-studied coupling agents responsible for synchrony in the SCN [1], [6], [7] along with its canonical receptor VPAC2R [8]. In mice, the abolition of VIP-based signaling by knockout of VPAC2R or VIP, or both, leads to weak rhythms or arrythmicity in behavior, clock protein oscillations, and firing rate in SCN explants [1]. The rhythmicity in a VIP knockout SCN explant can be restored by grafting a normal donor (wild-type) SCN via only paracrine VIP signaling [9]. The down-stream targets of VIP binding to VPAC2R acting via G-protein coupled receptor pathways involving the second-messengers cAMP and are activators of the period clock genes (Per1, Per2) [10], and VIP pulses can entrain SCN explants much like periodic light stimuli in mice [11]. VPAC2R has been also shown to be intracellularly expressed in a circadian manner, with the expression peaking in the early morning [12]. While the circadian expression/release of VIP is expected, in vivo VIP levels have not been conclusively measured [13]–[18]. A complete understanding of this critical synchronization phenomenon in the SCN remains an open question. In experimental and computational studies, coupling individual oscillators can achieve this robust synchrony, despite heterogeneity and different network connectivities. But, much less is known regarding the mechanisms and circuits involved in achieving this coupling, both due to system complexity and experimental limitations (as described above for VIP-VPAC2R coupling). Limiting the scope of possible coupling mechanisms that can achieve synchronization would have significant scientific value. Ueda et al. [19] and Korenčič et al. [20] identified the interactions between essential clock components (termed ‘core oscillator’) using only conserved transcriptional elements and phase relationships between core components, and thus outlined design principles for robust transcriptional oscillations. Similarly, in this work, we identify design principles behind VIP-VPAC2R-mediated coupling for robust synchrony, given the transcriptional elements and generic mode of VIP-VPAC2R coupling (assuming VIP-VPAC2R is the primary means of coupling). In particular, we computationally test the hypothesis that the phase of either VIP or VPAC2R, or both, relative to the core clock determines SCN synchrony and affects SCN properties, such as amplitude, period and entrainment range. We find that synchrony is only possible if VIP (an inducer of Per) acts in-phase with the transcriptional activity of Per promoters. Moreover, anti-phasic VIP action suppresses the network rhythms by moving the network into a desynchronous state, without completely suppressing individual oscillators. The modulation of afferent VIP signaling by phase-dependent VPAC2R expression [12] primarily improves the amplitude of SCN network rhythms and narrows the network entrainment range and has little effect on the ability of the network to synchronize. Finally, we have identified that an activating coupling agent, such as VIP, must follow a coupling-in-phase-with-promoters design principle to achieve robust synchrony, as an extension of the work of Ueda et al. [19]. Our findings are also applicable across several proposed models of the SCN network. Early computational models of the SCN focused on the fundamental biochemical mechanisms of the transcriptional-translational feedback loop (TTFL) that produces oscillations in the SCN [21], [22]. With the availability of real-time clock reporters and siRNA technology, detailed quantitative models of single SCN neuronal oscillators have been proposed [4], [20], [23]–[25]. Simultaneously, studies have considered properties of the SCN network vital to producing sustained and high precision oscillations: the role of strength of coupling [26]–[28], the role of network connectivity [29], [30], composition of damped and sustained neuronal oscillators [31], and modes of coupling [32]. These studies on coupling assumed a neuropeptide under clock control of sufficiently large amplitude to achieve synchrony. They focused, therefore, on the strength of coupling rather than on the effects of changes in the timing of coupling. The theory of coupled oscillators describes the role of phase of coupling in determining synchronous and asynchronous states of an oscillator network under the assumption of weak coupling between oscillators (see [33] and references therein). In the circadian realm, Indic et al. [34] show that an altered phase of coupling is necessary to split synchronized SCN oscillations into two anti-phase subgroups and thus explain the splitting behavior in hamsters using a theoretical formulation similar to ours. Ueda et al. [35] use a principle that synchronizing factors released during the day and night must have similar phase response curves to light- and dark-pulses, respectively, to identify potential coupling mechanisms in Drosophila. The work of Ueda et al. [35] is consistent with our theory and also suggests a role for the phase of these factors in synchrony. Our work is, to our knowledge, the first systematic investigation into the role of the phase (or timing) of neuropeptide coupling in determining properties of the SCN network, including synchrony and entrainment. We show that the phase of coupling is as important as and complements the amplitude of coupling in determining synchrony and entrainment across multiple models. Moreover, we explicitly consider the circadian control of the ligand (VIP) and receptor (VPAC2R) and highlight the asymmetry in the roles played by VIP and VPAC2R.
We first study the effect of the timing of VIP release alone on network synchrony and entrainment by setting VPAC2R expression to be constant. In addition to the VIP release time, we also varied the spread of intrinsic periods in the network in order to test the robustness of our results. The results are summarized in Figure 1. Time courses of the computational SCN reveal either synchrony (with VIP release at CT4) or desynchrony (at CT16) (Figure 1A). Synchrony is only achieved with VIP release in a small window (henceforth called the region of synchrony (ROS) ) within one circadian cycle (Figure 1B). The network always synchronizes for VIP release in the early subjective day (CT0-CT6), the time of peak intrinsic Per production, irrespective of the heterogeneity (period spread) of the network. Moreover, the system exhibits sharp transitions between regions of synchrony and desynchrony. This reflects a phase transition of the network into a synchronous state, a well-studied phenomenon in classical coupled oscillator theory [38]. This ‘cooperative’ process is nucleated by a small cluster of synchronized neurons that progressively recruit neighbors until the entire network is in synchrony. In particular, synchrony is attained when coupling-induced gene expression and intrinsic gene production in each neuron act synergistically, i. e. , coupling and intrinsic gene production are in-phase (for VIP release at CT4, Figure S1A). However, for VIP release at CT16, when the network does not synchronize, the coupling-induced expression and intrinsic production act antagonistically in each neuron (Figure S1A). The width of the ROS is dependent on the spread of intrinsic (uncoupled) periods of neurons in the network and the strength of coupling between neurons. The ROS is broadened (from CT0-CT8 to CT0-CT10) as the period spread in the network is decreased from its nominal value. Increasing the strength of coupling between neurons by over-expressing either VIP or VPAC2R broadens the ROS and can thus compensate for increases in period spread of the network (Figure S1B). While the network achieves perfect phase synchrony at low period spread with an SI of one, at the nominal spread, the phases of individual neurons are not perfectly aligned () (see Figure 1B). The snapshot of rhythms in individual oscillators in the synchronized network shows that neurons with nominal period spread are arranged in a linear phase pattern (Figure 1C). Within this pattern, some neurons peak earlier (phase advanced) than the network rhythm, while others peak later (phase delayed). Neurons with intrinsic period longer than the mean period are phase advanced and neurons with shorter intrinsic period are phase delayed (Figure 1C). The width of the linear phase ordering is significantly compressed as the period spread of the network is decreased (Figure S1C). Outside the ROS, while the neurons continue to oscillate (Figure 1A), the phases of the neurons are dispersed (Figure 1C). Although the neurons continue to interact, their ordering strongly resembles an uncoupled network of neurons as measured by the SI (Figure 1B). Despite the strong coupling signal between neurons in the network, the coupling maintains the system in a ‘uncoupled’-like state and prevents synchrony. An almost perfect positive correlation between individual neuron periods in the desynchronized network and their intrinsic periods further supports this view (not shown). Thus, the coupling might be viewed as forcing desynchrony in the neuronal network. The period of the Per gene network rhythms is close to the average intrinsic period of individual neurons in the network (Figures S1D and S1F). The period of the network rhythm is modulated by the phase of VIP release within the ROS. The amplitude of the network rhythms is boosted by synchrony within the ROS as measured by both absolute amplitude and amplitude relative to the mean Per expression (Figures S1E and S1G). While individual neuron amplitudes without VIP-VPAC2R signaling is, coupling reduces individual neuron amplitudes irrespective of the VIP release phase. Individual neurons in the coupled network have smaller amplitudes outside the ROS as compared to within the ROS. In other words, there is a significantly higher amplitude reduction (penalty) in desynchrony than in synchrony. The network entrains to sufficiently-strong external light-dark cycles (Zeitgeber) both within the ROS (at CT4) and outside the ROS (at CT16) (Figure S2A). Thus, the phase of VIP release does not affect the ability of the network to entrain to a Zeitgeber. However, the network shows different entrainment characteristics with VIP release at CT4 and at CT16. A synchronized SCN network (VIP release at CT4) behaves as a ‘strong’ oscillator that shows a narrow entrainment range and a phase angle of entrainment highly sensitive to the Zeitgeber period (see [39], [40] for definition of ‘strong’ and ‘weak’ oscillators). On the other hand, an SCN under desynchrony (VIP release at CT16) behaves as a ‘weak’ oscillator that exhibits a wide entrainment range and a phase angle of entrainment less sensitive to Zeitgeber period. We now further introduce circadian VPAC2R expression into the model. We study both the effect of the timing of VPAC2R expression and the strength of circadian modulation of VPAC2R when VIP is released at either CT4 or CT16. VPAC2R expression is under Per TTFL control and its strength is varied, while keeping the mean VPAC2R expression a constant, as shown in Figure 2A. Surprisingly, the ability of the network to synchronize is unaffected by the peak phase of VPAC2R expression (Figure 2B). As before, the SCN network can synchronize only with VIP release in the early subjective day (CT4), but not in the subjective night (CT16). The phase of VPAC2R expression affects, nevertheless, the amplitude of both individual oscillators in the network and the network rhythm (Figure 2C). VPAC2R peak expression in the early subjective day (coinciding with the ROS for VIP release) significantly improves the amplitude of individual oscillators and the network rhythm relative to constant VPAC2R expression. This boost in the network amplitude is caused by an increase in individual rhythm amplitude, since the synchronization index remains unchanged for different VPAC2R phases. VPAC2R expression in the subjective night suppresses the network rhythm and individual amplitudes. We also observed modulation of the network period by the phase of VPAC2R expression (Figure S3A). Next, we vary the strength of VPAC2R modulation between 0, meaning constant VPAC2R expression, and 1, meaning maximal circadian VPAC2R expression with no basal component (see Figure 2A). The amplitude of the network rhythm initially increases and then starts to decrease with increased strength of VPAC2R modulation for VPAC2R expression peak at CT4 (Figure 2D). Recall that VPAC2R expression at CT4 enhances network rhythm amplitude by boosting individual amplitudes (Figure 2C). On the other hand, with VPAC2R expression at CT16, where VPAC2R modulation reduces network amplitudes, increasing the strength of VPAC2R expression reduces amplitudes even further. However, while increasing strength of VPAC2R expression at CT16 reduces amplitudes, it results in period synchrony over a larger range of VPAC2R expression strengths than with VPAC2R expression at CT4. Similarly, the network period increases and decreases with changes in the strength of VPAC2R expression with VPAC2R peak at CT4 and CT16, respectively (Figure S3B). The entrainment properties of the network are also modulated by the phase of VPAC2R expression irrespective of the phase of VIP release (Figure S2B). Peak VPAC2R expression in the subjective night (at CT16) leads to an wider entrainment range relative to constitutive VPAC2R expression (Figure S2A) in both a synchronized (VIP release at CT4) and desynchronized (VIP release at CT16) SCN network. In other words, VPAC2R expression at CT16 makes the network behave as a ‘weaker’ oscillator than with constitutive VPAC2R expression [39]. On the other hand, while peak VPAC2R expression in the early subjective day (CT4) results in a much narrower entrainment range or a ‘stronger’ oscillator, peak VPAC2R expression at an intermediate phase (CT10) leads to an intermediate network entrainment range. Thus, VPAC2R expression phase can smoothly tune the entrainment range of the SCN network and the network' s qualitative behavior as a ‘strong’ or ‘weak’ oscillator. Finally, we test the robustness of our previous results on the network topology in the proposed model and the choice of model. We studied first a randomized network topology, since in the SCN only 10% of SCN cells express VIP [41] and only 30% of VIP cells express VPAC2R [42] making the network connectivity very heterogeneous. Replacing the all-to-all network connection with a random network topology or a local ring topology has little effect on the qualitative conclusions on synchrony and the timing of VIP release (Figures 3A and 3B). The change in topology and percentage connectivity of each neuron only alters the width of the ROS. Our conclusions regarding forced desynchrony, when the activating coupling agent is not in-phase with promoter of Per production, also remain unchanged. The qualitative observation that VIP release only in the early subjective day leads to synchrony and a lack of effect of the peak phase of VPAC2R expression on network synchrony appears to be robust to the choice of model as well. In the Goodwin-like model of Gonze et al. [26] defined in (7), the ROS for constitutive VPAC2R expression is restricted to a small window between CT2-4 (Figure 3C). As before, the poor synchrony in the subjective night is driven by desynchrony in the neurons as evidenced by robust individual neuron oscillation amplitudes, but very small network rhythm amplitudes (Figure S4A). As before, with oscillatory VPAC2R expression, the network remains in a state of synchrony (VIP release at CT2) or desynchrony (VIP release CT14) irrespective of the peak phase of VPAC2R expression (Figure 3D). However, as before, the phase of VPAC2R expression modulates the network amplitude and period (Figure S4B). One feature shared by both our DDE model and the Goodwin-like Gonze model is that VIP-VPAC2R coupling acts independently of the endogenous Per gene transcription according to an ‘OR’-logic gate (discussed in the Models section). While this is consistent with experimental data on the Per promoter [43], several prior models [26], [28], [29], [31] assume a dependent ‘AND’-logic gate for the coupling-driven and endogenous gene transcription. We verified using a variant of our DDE model (5) that our results were insensitive to the precise mechanism of coupling-driven and endogenous Per gene transcription (Figure S4C).
Our first key result is that VIP-based coupling must have a peak phase in the early subjective day to ensure synchronization in the SCN network. In showing this, we assumed that some combination of VIP production and release is under circadian control resulting in periodic VIP-based coupling (ignoring effects of the light environment). The experimental evidence regarding this assumption is mixed and sometimes inconsistent in mice and rats, two closely-related model species. In rats, overwhelming evidence suggests that both VIP mRNA expression and protein levels do not oscillate in constant darkness (DD condition) and oscillate only under light-dark (LD) conditions [15], [17], [44], [45]. However, VIP released into the extracellular medium in cultured rat SCN is circadian [18]. In mice, Dardente et al. [13] and Laemle et al. [16] show that VIP mRNA and protein levels in the SCN are circadian even under DD conditions. Although the circadian nature of VIP release is debated, we can study its effects via computational modeling. We identified an important design principle for coupling in the SCN: for activating coupling agents, the inducing activity of the coupling agent must be in-phase with the endogenous activity of the target gene' s promoters. This result is aided by our choice of a minimalist delay-based model of the SCN, and the generality of this principle across other choices of SCN model and network connectivity further strengthens our assertion. In fact, this principle is a generalization of prior work on the phasing of transcriptional regulation in single oscillators [19], [20] to a network of oscillators. Ueda et al. [19] and Korenčič et al. [20] both show that activating transcription factors should be in-phase, and suppressing transcriptions factors must be anti-phase with the activating transcription factors to achieve large amplitude. Clearly, neurons with improved amplitude from optimal inter-neuronal coupling can better influence and align wayward neurons back into a stable relationship. We also expect therefore that coupling agents that suppress core clock transcription or enhance degradation to follow the anti-phase with intrinsic gene production design principle, although we do not have a biological example for this. Antiphase VIP action results in desynchrony among robustly oscillating neurons resembling adult SCN in mice [46]. Moreover, SCN explants from neonates are synchronous and lose synchrony only during development. Our theory on timing of coupling suggests an explanation for this effect, since the phase of VIP action has been measured to change by several hours with age [44]. Recent work on another coupling factor, GABA, has shown that it works against synchrony under steady-state conditions [47], but acts to resynchronize the SCN under anti-phase configurations [48]. These observations are consistent with the hypothesis that GABA signaling endogenously occurs at the ‘incorrect phase’ (out-of-phase with repressors, since GABA is an inhibitory neuropeptide). Thus, GABA normally reduces synchrony among neurons, and only when neurons of the dorsal and ventral SCN are forced out-of-phase under extreme photoperiods [48], GABA behaves as a synchronizing neuropeptide. This role reversal of GABA with the phase configuration of the SCN is similar to the change in behavior of VIP from a synchronizer to a desynchronizer with overexpression of the neuropeptide [49] and supports our view that the phase and strength are complementary features of coupling. Entrainment of the network to an external Zeitgeber revealed differences in its behavior under synchronizing and desynchronizing VIP action. A synchronized network behaves as a ‘strong’ oscillator, while a desynchronized network behaves as a ‘weak’ oscillator. Comparing a coupled SCN network and uncoupled lung tissue, Abraham et al. [39] show by measuring entrainment ranges that the strength of coupling determines the ‘strong’ or ‘weak’ nature of oscillators. Our results reveal that even for the same strength of coupling, the phase of coupling determines the entrainment behavior of the SCN oscillator. This complementarity between coupling strength and phase also applies to recent observations that weakened coupling by means of vasopressin receptor knockouts speeds up jet-lag shifts in mice [50]. Simulations of 8 h jet-lag advances and 8 h delays showed faster re-entrainment with anti-phase VIP action (i. e. , in a desynchronized SCN) (Figure S2C). These results are also consistent with the recent finding that desynchronizing the network with VIP prior to a jet-lag protocol leads to faster re-entrainment after the jet-lag shift [49]. Another general feature of this coupling is the asymmetric effect of circadian control of the neuropeptide and its receptor on synchrony in the network. Synchrony in a network requires exchange of phase information between the constituent oscillators. Circadian release of neuropeptide informs the network of the phase of the releasing neuron aiding consensus in-phase and ultimately synchrony. However, the phase of the receptor merely modulates the effect of the rest of the network on the neuron. Thus, circadian neuropeptide release is globally informative, while circadian receptor expression is only locally informative. Next, we assemble available literature data on the timing of different aspects of VIP-VPAC2R signaling. Laemle et al. [16] show VIP immunoreactivity within SCN neurons peaking in the early subjective day and late subjective night entirely consistent with our prediction. Dardente et al. [13] estimate peak VIP mRNA expression at around CT18 allowing VIP release to occur in the early subjective day after delays in translation and extracellular release. While Shinohara et al. [18] show VIP release in multiple rat SCN explants in a range between CT16-CT24, attribution of CTs is made difficult by significantly shortened period (approx. 21 h) of VIP oscillations. Moreover, these measurements come from neonatal SCNs and there is some evidence that VIP mRNA phases change by up to 12 hours during development [44]. The available data on VIP timing are thus mostly consistent with our prediction of VIP release in the early subjective day. Information on second-messengers and transcription factors in the G-protein coupled receptor-mediated VIP-VPAC2R signaling to the core circadian clock [7] can also be used to test our predictions indirectly. Brancaccio et al. [51] and Enoki et al. [52] both show using real-time reporters that intracellular oscillates and peaks at CT6-7. Moreover, cAMP, the other important second-messenger, also peaks in the early to mid subjective day (CT6) as reported by O' Neill et al. [53]. Finally, VIP-VPAC2R signaling converges on the core circadian clock through the transcription factor CREB. CREB-mediated activation of Per gene expression occurs in the mid-subjective day (CT6) [54]. Although this evidence reinforces our claim that coupling must occur within a certain circadian time window in the subjective day, the particular CT of action is likely to be determined by other factors including entrainment, period control and SCN heterogeneity. The most direct measurements of VPAC2R expression under DD conditions comes from An et al. [12] showing peak bioluminescent staining of SCN slices at subjective dawn (CT0). Since we expect synchrony with robust amplitude with VIP in-phase with VPAC2R in the early subjective day, the study reinforces both our claims regarding phase of VIP release and VPAC2R expression. Moreover, VPAC2R mRNA expression is found to either peak around dawn (ZT0) [15] under LD conditions, or be circadian, but biphasic, in DD and LD conditions [55]. However, recent work on post-transcriptional control [56] has shown that having circadian mRNA expression is no guarantee of circadian protein levels and vice versa. Circadian VPAC2R expression did not have a significant effect on synchrony in our model, but could tune the amplitude and period of the network. We also observed that the phase of circadian VPAC2R alters the entrainment behavior of the network by modulating the network entrainment range. That VPAC2R expression in the early subjective day narrows the entrainment range is expected, since under similar VPAC2R expression phase we observe increased network amplitude. As noted in [39], changes in network amplitude and period indeed affect the entrainment properties of the network oscillator. Although we identify several critical network properties modulated by circadian VPAC2R expression, we cannot resolve the specific role of the circadian VPAC2R expression that is observed [12]. Evidence in the literature cited above support our claims, but our predictions could be more directly tested. An experiment that significantly alters the phase of VIP release in an SCN explant would be expected to desynchronize the individual rhythms and damp out network rhythms. It would, for example, be interesting to test the effects of putting the Vip gene under the control of the Bmal1 promoter. This would, in principle, put VIP production in anti-phase to its normal physiological release time. A more easily-accessible phenotype of alterations in the phase of VIP-VPAC2R coupling is the period of an explant or behavioral rhythms. The period of network rhythms is affected by both phase of VIP release and VPAC2R expression in our model (Figures S1C, S2E and S3A). Moreover, the phase angle of entrainment would also change with the change in period. These experiments could follow the approach of previous studies on VIP and VPAC2R signaling [6], [42]. While we focused on the VIP-VPAC2R coupling mechanism, our results are equally applicable to other coupling neuropeptides and their receptors, such as vasopressin (AVP) and its receptor (AVPR) or gastrin-releasing peptide (GRP) and its receptor in mammals. Our work also raises interesting possibilities for multiple coupling neuropeptides with different strengths and phases operating synergistically to synchronize and entrain the SCN under different physiological and environmental conditions, such as with GABA and VIP observed in [48]. We did not, however, take into account the heterogeneity in the expression of these different neuropeptides and their receptors. VIP is mainly expressed in the core region, where oscillators appear to be weak or damped, while AVP is expressed predominantly in the shell with robust oscillations [57]. VIP is expressed in 10% of SCN neurons [41], up to 90% of neurons express the VPAC2R [12], thereby allowing these neurons to respond to VIP signaling. While VPAC2R is expressed in only 30% of VIP cells, it is expressed in 50% of AVP cells [42]. So the core VIP neurons seem to signal a different class of shell AVP neurons. Thus, there is a complex topology of interactions and core clock properties [58] that we have not accounted for. Since our results are based on the interaction of coupling with a core clock mechanism, they are qualitatively independent of these specific neuronal and network properties and could be applied to other functionally-similar neuropeptides in other organisms, such as pigment dispersing factor (PDF) in Drosophia [7], [59].
We test the robustness of our results (obtained from the DDE formulation (4) ) to the choice of model by running simulations using an alternate circadian core model – the one of Gonze et al. [26]. In the modified Gonze et al. model, the circadian oscillator in neuron with clock gene mRNA, clock protein and transcriptional inhibitor in a network with connectivity is governed by: (7) with parameters. Equation (7) corresponds to the original DDE system in (4) with phase control of both VIP and VPAC2R oscillations by and respectively.
Synchronization among multiple oscillators is a common theme in many biological and engineered systems. Here, we look at its use by the mammalian biological clock to keep accurate time. Through biochemical interactions among a network of inaccurate neuron clocks, a strong precise clock is produced. Although we are gradually learning more about these biochemical interactions, the details still remain largely unclear. Studies, both computational and experimental, have shown that the strength of the rhythmic interaction critically decides if a system can synchronize, i. e. , the interactions must be strong enough. In this work, we show that the rhythmic interaction between these neuronal clocks must be timed correctly (in the right phase) in addition to being strong enough to synchronize the network. Activating (repressing) interactions must coincide with other activators (repressors) in each neuron to achieve synchrony. Since this principle imposes certain properties on synchronizing interactions, these properties can be used to identify and understand novel interaction mechanisms. Further, these principles are applicable to interactions between cellular oscillators in other tissues and organisms.
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Timing of Neuropeptide Coupling Determines Synchrony and Entrainment in the Mammalian Circadian Clock