Stochastic Simulation of DelayInduced Circadian Rhythms in Drosophila
 Zhouyi Xu^{1} and
 Xiaodong Cai^{1}Email author
DOI: 10.1155/2009/386853
© Z. Xu and X. Cai. 2009
Received: 11 December 2008
Accepted: 10 May 2009
Published: 15 June 2009
Abstract
Circadian rhythms are ubiquitous in all eukaryotes and some prokaryotes. Several computational models with or without time delays have been developed for circadian rhythms. Exact stochastic simulations have been carried out for several models without time delays, but no exact stochastic simulation has been done for models with delays. In this paper, we proposed a detailed and a reduced stochastic model with delays for circadian rhythms in Drosophila based on two deterministic models of Smolen et al. and employed exact stochastic simulation to simulate circadian oscillations. Our simulations showed that both models can produce sustained oscillations and that the oscillation is robust to noise in the sense that there is very little variability in oscillation period although there are significant random fluctuations in oscillation peeks. Moreover, although average time delays are essential to simulation of oscillation, random changes in time delays within certain range around fixed average time delay cause little variability in the oscillation period. Our simulation results also showed that both models are robust to parameter variations and that oscillation can be entrained by light/dark circles. Our simulations further demonstrated that within a reasonable range around the experimental result, the rates that dclock and per promoters switch back and forth between activated and repressed sites have little impact on oscillation period.
1. Introduction
Almost all living organisms, including animals, plants, fungi, and cyanobacteria, exhibit daily periodic oscillations in their biochemical or physiological behavior, which are known as circadian rhythms [1–7]. The mechanism of circadian oscillation has been an extensive research topic in the last three decades. It has been found that circadian rhythms in fact are determined by oscillatory expression of certain genes [89]. Specifically, circadian clocks consist of a network of interlocked transcriptionaltranslational feedback loops formed by a number of genes [2]. In Drosophila, transcription of per and tim genes is activated by a heterodimer consisting of two transcriptional activators dCLOCK and CYCLE [10–13]. The PER protein in turn binds to the dCLOCKCYCLE heterodimer, which inhibits the DNA binding activity of the dimer, thereby repressing the transcription of per and tim [11–14]. While this forms a negative feedback loop, there is also a positive feedback loop, in which PER and TIM activate dCLOCK synthesis by binding dCLOCK and relieving dCLOCK's repression of dclock transcription [1516].
Several mathematical models have been proposed for circadian oscillation in Drosophila [121417–22]. The models of Smolen et al. [1214] introduce time delays in the expression of dclock and per genes, while other models do not have such delays. Numerical simulations using ordinary differential equations (ODE) show that all these models can produce circadian oscillations. In particular, times delays were found to be essential for simulation of circadian oscillations with the model of Smolen et al. [1214].
Since there is significant stochasticity in gene expression arising from fluctuations in transcription and translation [23–25], it is desirable to simulate circadian oscillations in the presence of noise. Toward this end, several stochastic models were proposed [426–29], and Gillespie's stochastic simulation algorithm (SSA) [3031] was employed to simulate circadian oscillations. All these stochastic models [426–29] do not include time delays. In order to reflect the noise in gene expression, Smolen et al. used two approximate stochastic simulation methods to simulate circadian oscillation based on their models with delays [1214]. However, their models lumped transcription and translation into one single process and did not model the process that dCLOCK binds to or dissociates with dclock and per genes to activate or inhibit transcription. Since transcription is a major source of intrinsic noise [2324], the approximate stochastic simulation of Smolen et al. may underestimate the effect of noise. Li and Lang [32] used similar approximate stochastic simulation methods to simulate reduced model of Smolen et al. [14], but with an emphasis on the noisesustained oscillation in the region of parameter values where the deterministic model predicted no oscillation. Currently, no exact stochastic simulation has been done for circadian rhythm models with random delays, partially due to the fact that Gillespie's SSA cannot handle delays in certain reactions.
Recently, we developed an exact SSA algorithm for systems of chemical reactions with delays [33]. The goal of this paper is to apply this exact SSA to simulate circadian oscillations in Drosophila using a model with time delays and to investigate the effects of noise and random time delays on circadian oscillations. We first develop two stochastic models with random delays for circadian oscillations in Drosophila based on the two deterministic models of Smolen et al. [1214]. Using our exact SSA, we then simulate freerunning circadian oscillation under constant darkness. Our simulations demonstrate that both models can produce sustained oscillations. The variability in oscillation period is very small although the variability in oscillation peaks is considerably large. In particular, although time delays are essential to oscillation, random fluctuations in time delays do not cause significant changes in oscillation period as long as the average delays are fixed. Our simulations also showed that circadian oscillations of both models are robust to parameter variations. The entrainment by light was also simulated for both models, yielding results consistent with experimental observations. To see the effect of transcription noise, we also run simulations with different values for the rate that dCLOCK binds or unbinds to per and dclock genes.
2. Methods
2.1. The Detailed Model of Circadian Oscillation with Time Delays
2.1.1. Model Description
Detailed stochastic model for the circadian oscillator.
Rid  Reaction  Rate ( ) 

1  dclockg dclockg + dclockm 

2  dclockm 

3  dclockm dclockm + dCLOCK (delay ) 

4  dCLOCK 

5  dclockg + dCLOCK dclockg.dCLOCK 

6  dclockg.dCLOCK dclockg + dCLOCK 

7  dCLOCK + perg perg.dCLOCK 

8  perg.dCLOCK dCLOCK + perg 

9  perg.dCLOCK perg.dCLOCK + perm 

10  perm 

11  perm perm + (delay ) 

 , 

22 


 dCLOCK + dCLOCK. 

 dCLOCK. dCLOCK. , 

 dCLOCK. , 

54  dCLOCK. 

Molecular species.
Species  Description 

dclockg  gene 
dclockm  mRNA 
dCLOCK  dCLOCK protein 
dclock.dCLOCK  gene bounded by dCLOCK protein 
perg  gene 
per.dCLOCK  gene bounded by dCLOCK protein 
perm  mRNA 
,  PER protein with phosphorylations 
dCLOCK. ,  Complex of dCLOCK and PER with phosphorylations 
Reaction represent transcription of gene, degradation of mRNA, translation of mRNA, and degradation of dCLOCK protein, respectively. Reaction 5 models the process that dCLOCK protein binds to gene and reaction 6 represents dissociation of dCLOCK with . Reaction 7 and 8 specify the event that dCLOCK binds to and dissociates with gene. Reactions 9, 10, and 11 represent transcription of gene after it is activated by dCLOCK, degradation of mRNA, and translation of mRNA, respectively. Reactions represent the phosphorylation of PER and reaction 22 represents the degradation of PER. Reactions represent the association of dCLOCK with PER at different levels of phosphorylation. Reactions describe the phosphorylation of dCLOCK and ( ) heterodimer. Reactions represent the degradation of dCLOCK and ( ) heterodimer.
2.1.2. Parameter Estimation
Each reaction is associated with a reaction probability rate constant, , which determines the probability that a specific reaction occurs in an infinitesimal time interval. The probability rate constant of a specific reaction can be calculated from conventional rate constant as follows: for a monomolecular reaction, for a bimolecular reaction with two different reactants, and for a bimolecular reaction with one reactant [39], where , and is the Avogadro constant and is the system volume. We assume that a lateral neuron in is a sphere of a radius around 6 [1440], which results in a volume L. As many other existing models [121441], we do not separate nuclear and cytoplasmic compartments. We retain most parameter values from Smolen's et al. [12] including and . The remaining 10 parameters, , , , and , are determined in our simulation. In the following, we describe 54 reactions and how each probability rate constant was determined.
Here represents the concentration of the species in the bracket. A time delay is included in reaction 3 accounting for time needed for transcription, translation, and other potential mechanisms for activating the transcription of . Smolen et al. chose to be a deterministic number equal to 5 hours [12]. Taking into account uncertainty in this delay, we choose as a random variable uniformly distributed in the interval (4h–6h).
In reaction 5, dCLOCK binds to the Ebox of [1113], but there is no experimental report on the values of and the dissociation rate . However, the dissociation rate of myogenin protein with the Ebox of E12 gene was reported to be [44]. Therefore, we choose . The equilibrium constant of reactions 5 and 6 is equal to the Michaelis constant in [12] that describes the regulation of dCLOCK synthesis by dCLOCK and was chosen to be 1 nM [12]. Using this equilibrium constant, we calculate to be . Reactions 7 and 8 specify the event that dCLOCK binds to and dissociates with gene. The equilibrium constant of reactions 7 and 8 is equal to the Michaelis constant in [12], which was 1 nM. This Michaelis constant reflects the regulation of PER synthesis by dCLOCK. After choosing , is calculated from the equilibrium constant as .
The transcription rate of gene is chosen to be 20 , and the degradation rate of mRNA is calculated as from the halflife of mRNA which was estimated to be 2 hours [4243]. Also, we assume that there are two copies of gene, and thus, the initial value for the number of molecules of perg in Table 1 is 2. The synthesis rate of PER protein was chosen to be nM in the model of Smolen et al. [12], which equivalently is molecules per hour. In our model, the average PER synthesis rate is molecules per hour. Letting , we got . Similar to , we also tested the sensitivity of simulation results to . Increasing or decreasing two times while fixing the ratio of only causes negligible change in the mean and standard error (SE) of period and peaks (data not shown).
A delay is introduced in reaction 11. This time delay accounts for the time needed for the transcription and translation of gene. Smolen et al. [12] chose to be 8 hours. However, the total time needed for transcription and translation maybe less than 8 hours [45] and also there may be some fluctuations in . Therefore, we chosen as a random variable with mean 6 hours, uniformly distributed in (4.8h–7.2h).
Reaction 22 represents the degradation of PER and is equal to , where nM and nM [12].
Reactions represent the association of dCLOCK with PER at different levels of phosphorylation. The deterministic rates for all these reactions are 30 n and thus the probability rate constants are , . Reactions describe the phosphorylation of dCLOCK and ( ) heterodimer and we have , . Reactions represent the degradation of dCLOCK and ( ) heterodimer. We have , and , where and are given earlier.
2.2. The Reduced Model of Circadian Oscillation with Time Delays
2.2.1. Model Description
Reduced stochastic model for the circadian oscillator.
Rid  Reaction  Rate ( ) 

1  dclockg dclockg + dclockm 

2  dclockm 

3  dclockm dclockm + dCLOCK (delay ) 

4  dCLOCK 

5  dclockg + dCLOCK dclockg.dCLOCK 

6  dclockg.dCLOCK dclockg + dCLOCK 

7  dCLOCK + perg perg.dCLOCK 

8  perg.dCLOCK dCLOCK + perg 

9  perg.dCLOCK perg.dCLOCK + perm 

10  perm 

11  perm perm + PER (delay ) 

12  PER 

13  dCLOCK + PER dCLOCK.PER 

14  dCLOCK.PER 

2.2.2. Parameter Estimation
The rate is chosen to be 10 which is slightly lower than that in the detailed model. This is because the synthesis rate of dCLOCK protein in the reduced model of Smolen et al. [14] was nM , which is smaller than in the detailed model. The rate is the same as that in the detailed model. Letting equal to , we calculate . The rate is the same as that in the reduced model of Smolen et al. [14], equal to 0.5 .
The unbinding rate of dCLOCK to gene, , is chosen identical to that in the detailed model. The equilibrium constant , which is equal to the Michaelis constant in [14] that describes the regulation of dCLOCK synthesis by dCLOCK, was chosen to be 0.1 nM [14]. Using this equilibrium constant, we calculate to be 1.44 . Similarly, is the same as that in the detailed model. The equilibrium constant , which is equal to the Michaelis constant in [14] that describes the regulation of PER synthesis by the transcriptional activators dCLOCK, is chosen to be 0.3 nM [14]. Then we calculate to be 0.48 .
The transcription rate of gene is chosen to be 10 , which is lower than that in the detailed model, because the synthesis rate of PER protein in the reduced model of Smolen et al. [14] was nM which is smaller than that in the detailed model. The degradation rate of mRNA is again . Letting equal to , we calculate as . The degradation rate of PER is the same as that in the reduced model of Smolen et al. [14], equal to 0.5 . The degradation rate of dCLOCK and PER complex is , identical to that in the detailed model and we have .
Time delays and are chosen as follows. As the effective delay contributed by PER phosphorylation is incorporated into and , , and should be longer than those in the detailed model. Therefore we chose and uniformly distributed in the time interval (5h–9h) and (7h–11h), respectively.
2.3. Stochastic Simulation
Gillespie's SSA [31] is often employed to simulate the stochastic dynamics of genetic networks [2346]. However, Gillespie's SSA cannot deal with delays in certain reactions. Recently, we developed an exact SSA for systems of chemical reactions with delays [33], which can handle both deterministic and random delays. We use this exact SSA to simulate the dynamics of the systems described in Tables 1 and 3.
2.4. Data Analysis
Customized MATLAB Software (Mathworks Inc.) was written to analyze data generated from stochastic simulations, for example, to calculate the mean and SE of protein levels, to identify the peaks of dCLOCK and PER during oscillation, and to calculate the peak amplitudes. Oscillation periods were calculated using the shorttime Fourier transform (STFT) method [47]. Specifically, Fourier transform was applied to protein levels of dCLOCK and PER within a time window of 70 hours, after the mean level was subtracted. The largest peak at a nonzero frequency was identified as the oscillation frequency within the time window and the period of the oscillation is the inverse of the oscillation frequency. Note that the maximum period that can be identified by the STFT is 35 hours since a time window of 70 hours was used.
3. Results
3.1. Simulation of Oscillation in the Presence of Noise
Statistics of oscillations for the detailed stochastic model.
Mean  SE  CV  

Period (h)  23.93  0.78  3.26% 
Peak value of total PER  10149  1530.1  15.08% 
Peak value of free dCLOCK  1377.1  184.62  13.41% 
Peak value of total dCLOCK  1437.1  156.24  10.87% 
Peaktothrough amplitude of total dCLOCK  1016.5  191.51  18.84% 
Statistics of oscillations for the reduced stochastic model.
Mean  SE  CV  

Period (h)  23.60  0.80  3.39% 
Peak value of Total PER  196.47  40.49  20.61% 
Peak value of free dCLOCK  183.09  35.49  19.38% 
Peak value of total dCLOCK  201.51  31.05  15.41% 
Peaktothrough amplitude of total dCLOCK  172.47  36.10  20.93% 
We investigate the effect of the random time delays with fixed average time delays. Since both detailed and reduced models produced similar results, we here only present results for detailed model. Note that time delays and in our simulations are random variables uniformly distributed in , where the standard value of and standard value of , with denoting the average time delay. To test the sensitivity of the range of random delays, we run more simulations using different and but with a fixed . Specifically, when we fix to be 5 hours and 6 hours for and , respectively, if and are uniformly distributed in , the mean period is 23.89 hours and the SE is 0.76 hour; if and are uniformly distributed in , the mean period is 23.81 hours and the SE is 0.79 hour. In both cases, the mean period and SE are very close to the results from standard value of and . Therefore, our simulations show that the random changes in the delays do not cause significant variations in the oscillation period as long as the average delays are fixed.
Smolen et al. [1214] also investigated the effects of noise using stochastic simulation. There are three major differences between our stochastic simulation and that of Smolen et al.: ( ) we employed exact SSA, whereas they used approximate SSAs, ( ) two delays critical to circadian oscillation are random in our simulation but deterministic in the simulation of Smolen et al., and ( ) we explicitly simulated the transcription process and the binding/unbinding events between dCOLCK and per and dclock promoters, whereas Smolen et al. lumped transcription and translation of dclock and per into a onestep process.
To convert concentration into number of molecules, we used the volume of typical lateral neuron cells, whereas Smolen et al. determined a scale factor by trial. For the detailed model, this resulted in different scale factors and protein levels in our simulation as shown in Figure 2 are approximate 10 times of those in the simulation of Smolen et al. as depicted in Figure 3 of [12]. To make fair comparison, we ran simulations using the same scale factor as Smolen et al. [12]. Our simulation results showed that the mean peak values of PER, free dCLOCK and total dCLOCK are 1205, 176, and 183, respectively, which are comparable to the results of Smolen et al. [12]. The mean period in our simulation is 24 hours and the CV of periods is 3.33%. These results are also comparable to the results of Smolen et al.: a mean period of 23.5 hours and a CV of 5%. The CVs of the peaks of PER, free dCLOCK, and dCLOCK in our simulation are 15.47%, 15.20%, and 12.26%, respectively, which are greater than the CV of PER (9%) in the simulation of Smolen et al. [12]. For the reduced model, it turns out that protein levels in our simulation are similar to those in the simulation of Smolen et al. [14]. The CV of periods in our simulation (3.39%) is slightly smaller than that obtained in simulation of Smolen et al. (4.78%). Since no result about the CV of peak protein levels was reported by Smolen et al., we cannot compare the CV of peak protein levels.
In summary, although the noise in our models may be stronger than that in the models of Smolen et al. due to the random delays, transcription process, and random activation and repression of the promoters of per and dclock, the CV of periods in our simulation is slightly smaller than that in the simulation of Smolen et al. [14]. This result indicates that approximate simulation may have yielded nonnegligible errors. It is difficult to evaluate the effect of such possible errors in the approximate method of Smolen et al. [14], but our simulation method is exact and can correctly capture the stochastic dynamics of the circadian rhythm. It seems that strong noise in our model is reflected in the peak protein levels because the CVs of peak protein levels in our detailed model are larger than those in the detailed model of Smolen et al. [12].
3.2. Robustness Test in the Presence of Noise
In living cells, biochemical parameters often vary significantly from cell to cell due to stochastic effects, even if the cells are genetically identical [50]. But circadian oscillations with close period are still withstood in or mammals. Therefore, a model of circadian rhythm should be robust in the sense that small parameter variations should not lead to large period variations. For the deterministic models, Smolen et al. [1214] have shown that circadian rhythm is robust when a parameter changes its value by . Here, we test if circadian rhythm is robust with respect to parameter changes in the presence of intrinsic noise. To test robustness, each parameter is decreased or increased by from the standard value, with all other parameters fixed at the standard values, and then the mean and SE of oscillation periods and peaks are determined from simulation results. Since and are random variables, we decrease or increase their mean values by .
We also tested the robustness of oscillation for the reduced model. The reduced model has 14 probability rate constants and 2 time delays. Therefore, 33 sets of simulations were run including the set with standard parameter values. Figure 6 plots relative change of the mean value of the period and peaks for the parameter sets with one changed parameter comparing with the standard parameter set and Figure 7 plots the CV of the period and peaks for all parameter sets. It is seen that the change is small in period but relatively large in peaks when a parameter changes. It is also seen that CV of the periods is very small for both models, in the interval . Therefore, the system is very robust to the parameter variation in oscillation period. Note that the CV of the peaks of the reduced model is larger than that of the detailed model. This is due to the fact that the reduced model has lower number of molecules in the system so that there is larger internal noise.
As in the detailed model, the period, the peak of the free dCLOCK and the peak of the total PER in the reduced model are most sensitive to , , and , respectively. Specifically, decreasing (increasing) the mean value of by 20% decreased (increased) the mean period by ( ) and the corresponding CV was ( ). Decreasing (increasing) by decreased (increased) the mean peak of free dCLOCK by ( ) and the corresponding CV was ( ). Decreasing (increasing) by decreased (increased) the mean peak of total PER by ( ) and the corresponding CV was ( ). Comparing the results of two models, it appears that both models have small changes in the mean period and relatively large changes in the mean peaks and that the reduced model has slightly larger CVs.
3.3. Light Entrainment of Oscillation in the Presence of Noise
Models of circadian rhythms must be able to maintain synchrony with environmental cycles to drive behavioral, physiological and metabolic outputs at appropriate time of day [7]. Circadian rhythms can be entrained by external cues, such as daily environmental cycles of light and temperature, but light is generally considered as the strongest and most pervasive factor. Therefore, the responses of the rhythm are often simulated by light pulses or light/dark (L/D) cycles [121851–53]. We first consider the detailed model. In , light induces to enhance the degradation of phosphorylated TIM [125455]. Since there is no separate variable for TIM in our model, the degradation of phosphorylated PER was induced to simulate the effect of light, as done by Smolen et al. [12]. Here the phosphorylated PER includes all unbounded and bounded PERs ( and dCLOCK. dCLOCK. ). The dCLOCK is released after the PER complex with dCLOCK is degraded by light and the degradation rate of all phosphorylated PER is 0.9 [12]. In addition, to keep the oscillation period, the maximum degradation rate of dCLOCK, , was reduced to 1.5 nM [12] and the probability rate constants and , in our stochastic model were reduced correspondingly.
3.4. Impact of Transcription Activation Rate
As we mentioned earlier, the rate that dCLOCK binds to and genes is unknown but was estimated in our simulation. The rate that dCLOCK dissociates with and genes is chosen to be equal to the experimentally reported dissociation rate of myogenin protein with the Ebox of E12 gene. In a deterministic model, these rates generally do not affect oscillation as long as their ratio is fixed. However, these rates may have significant effects on transcription noise even when their ratio is fixed [232456]. In the following, we change the value of while keep the ratio fixed to see whether the oscillation period changes. Since both detailed model and reduced models yield similar results, we only give results for the detailed model.
4. Discussion
We have presented a detailed and a reduced stochastic model for delayinduced circadian rhythm in based on the deterministic models of Smolen et al. [1214], and employed our recently developed exact stochastic simulation algorithm [33] to simulate the circadian rhythm. This work is unique since no exact stochastic simulation has been carried out for circadian rhythms based on a model with random time delays. As discussed in [33], several SSAs have been developed for reaction systems with delays [5758]. However, the algorithm in [57] and two algorithms in [58] are not exact. The work in [33] also proved that another heuristic algorithm in [58] is exact but requires more computation than the exact SSA in [33]. Since both algorithms are exact, they should produce the same statistical results. Another SSA for systems with delays was proposed in [59], but an approach similar to that in [57] was used, and thus, it is not exact either. Smolen et al. [1214], as well as Li and Lang [32], also simulated delayinduced circadian oscillation, but they used approximate stochastic simulation methods.
Our simulation results demonstrated that the intrinsic noise causes large fluctuations in oscillation peaks but very small fluctuations in oscillation period. This observation is seen in all simulations under different conditions, such as constant darkness and L/D cycles. Deterministic simulation cannot reveal this phenomena, since both period and peaks are constant. Our stochastic simulations also showed that circadian oscillation is robust in the presence of noise in the sense that noise has little effect on oscillation period although it can change oscillation peaks significantly. We also showed that random delays within certain range do not cause significant variations in the oscillation period as long as the average delays are fixed. To the best of our knowledge, these two results have not been observed in previous stochastic simulation of circadian rhythms. These two observations imply that circadian oscillation is robust in the presence of noise and random delays and that the randomness inherent to the oscillation circuit may not have much biological impact on the organism. As discussed in [50], when a protein regulates its targets, it often operates on a Hill curve. Once the level of the regulating protein is higher or lower than certain value, the protein operates at the top or bottom of the curve and the fluctuation of its level to certain extend does not affect much the regulating effect on its targets. Therefore, the relatively large variations in the peak values of PER and dCLOCK proteins observed in our simulation may not have a strong biological impact.
Similar to previous deterministic simulations and approximate stochastic simulations [1214], our stochastic simulation shows that both detailed and reduced stochastic models can provide sustained oscillations under darkness and L/D cycles. Our results also show right phase of all the components in the system, correct phase and antiphase relationship of mRNAs and proteins, and also the appropriate lags between mRNAs and proteins. Our stochastic simulation further demonstrated that circadian rhythm is robust to parameter variations in the presence of noise. Increasing or decreasing each parameter by of its standard value changes the mean period by less than and causes negligible changes in the CV of oscillation periods. The model is not sensitive to the time delay during the mRNA translation, but it is most sensitive to the average time delay during mRNA translation, which shows that time delay is essential to circadian oscillation in the two models. However, random fluctuations in these two time delays have little effect on the oscillation period as long as the average delays are fixed. We also found that the binding and unbinding rates of dCLOCK to and genes within a reasonable range have little impact on the circadian oscillation. Increasing or decreasing the binding and unbinding rates by 10 times relative to an experimentally reported rate while keeping their ratio fixed does not cause significant changes in the period and peaks of oscillation.
We have compared our exact simulations with approximate simulations of Smolen et al. [1214] in Section 3.1. Another work by Li and Lang [32] also employed approximate SSAs to simulate the reduced model of Smolen et al. [14]. Like Smolen et al. [1214], Li and Lang [32] used deterministic delays, whereas we employed random delays which are more appropriate to reflect the delays in transcription, translation, and other chemical process. Li and Lang emphasized on the noise induced oscillation and showed that noise can sustain oscillation in the parameter region where no oscillation is predicted by the deterministic model, whereas we here focused on the robustness of oscillation in the presence of intrinsic noise and the effect of random delays. We showed that the oscillation is robust in the presence of noise since there is very little variability in oscillation period in spite of large random variability in peaks, and that random changes in delays within a large interval around the fixed average delay cause little variability in the oscillation period.
Declarations
Acknowledgments
This work was supported by the National Science Foundation (NSF) under NSF CAREER Award no. 0746882.
Authors’ Affiliations
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