Bioinformatics Advance Access originally published online on May 11, 2006
Bioinformatics 2006 22(14):1775-1781; doi:10.1093/bioinformatics/btl182
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Synchronizing a multicellular system by external input: an artificial control strategy
1 Aihara Complexity Modelling Project, ERATO, JST 4-6-1 Komaba, Meguro, Tokyo 153-8505, Japan
2 Institute of Industrial Science, The University of Tokyo Bunkyo-Ku, Tokyo 113-8656, Japan
3 Department of Electrical Engineering and Electronics, Osaka Sangyo University Daito, Osaka 574-8530, Japan
*To whom correspondence should be addressed.
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ABSTRACT |
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 TOP ABSTRACT 1 INTRODUCTION2 THE MODEL3 RESULTS4 CONCLUSIONREFERENCES |
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Motivation: Although there are significant advances on elucidating the collective behaviors on biological organisms in recent years, the essential mechanisms by which the collective rhythms arise remain to be fully understood, and further how to synchronize multicellular networks by artificial control strategy has not yet been well explored.
Results: A control strategy is developed to synchronize gene regulatory networks in a multicellular system when spontaneous synchronization cannot be achieved. We first construct an impulsive control system to model the process of periodically injecting coupling substances with constant or random impulsive control amounts into the common extracellular medium, and further study its effects on the dynamics of individual cells. We derive the threshold of synchronization induced by the periodic substance input. Therefore, we can synchronize the multicellular network to a specific collective behavior by changing the frequency and amplitude of the periodic stimuli. Moreover, a two-stage scheme is proposed to facilitate the synchronization in this paper. We show that the presence of the external input may also initiate different dynamics. The multicellular network of coupled repressilators is used to show the effectiveness of the proposed method. The results not only provide a perspective to understand the interactions between external stimuli and intrinsic physiological rhythms, but also may lead to development of realistic artificial control strategy and medical therapy.
Availability:
Contact: aihara{at}sat.t.u-tokyo.ac.jp
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1 INTRODUCTION |
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TOPABSTRACT1 INTRODUCTION2 THE MODEL3 RESULTS4 CONCLUSIONREFERENCES |
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Cells are continuously subjected to conditions in the form of both intrinsic rhythms generated by intracellular clocks and extrinsic driving from extracellular environment (Garcia-Ojalvo et al., 2004; McMillen et al., 2002). The rhythm generators are composed of a lot of clock cells which are intrinsically diverse but nevertheless manage to function in a coherent oscillatory manner (Gonze et al., 2005). Physiological function derives from the interactions of the cells not only with each other but also with extracellular medium to generate rhythms essential for life. It seems likely that many bodily activities require synchronization of cellular ones (Glass, 2001). Relevant examples are the heartbeat generated by the sinoatrial node composed of thousands of pacemaker cells which interact with each other to set the normal cardiac rhythm (Wilders and Jongsma, 1993), the circadian clock residing at the suprachiasmatic nuclei (Yamaguchi et al., 2005) and synchronization as an essential component of many cortical functions (Gray et al., 1989). Although there are significant advances on elucidating the collective behaviors on biological organisms in recent years, the essential mechanisms by which the collective rhythms arise remain to be fully understood, and further how to synchronize multicellular networks by artificial control strategy has not yet been well explored.
External inputs are known to play an important role to synchronize biological rhythms. For instance, organisms usually display a circadian rhythm in which key processes show a 24 h periodicity entrained to the light–dark cycle (Goldbeter, 1996; Schultz and Kay, 2003). Other examples are physiological rhythms stimulated by regular or periodic inputs occurring in the context of medical devices (Simoin et al., 2000), synchronization of electronic genetic networks by an external forcing, i.e. external voltage (Wagemakers et al., 2006), and a wide variety of regular and irregular rhythms induced by periodic stimulation of a squid giant axons (Kaplan et al., 1996; Matsumoto et al., 1987). In general, physiological oscillations can be synchronized to appropriate external or internal stimuli. However, even simple models may show enormous complexity that arises from periodic stimulation of nonlinear oscillations. For example, the complexity arises from continuously periodic stimulation of Duffing's system (Lai et al., 2005), continuously (Tang and Chen, 2003) or impulsively (Liu et al., 2005) periodic stimulation of predator–prey models. Therefore, it is important to analyze the effects of external stimuli on intrinsic physiological rhythms, and better understanding of the interactions between the stimuli and physiological rhythms may lead to the development of control strategy and medical devices.
Many different mathematical models have been proposed to analyze collective rhythms in diverse systems. For example, synchronization has been observed in mathematical models of coupled repressilators (Garcia-Ojalvo et al., 2004; Wang and Chen, 2005), relaxation oscillators (McMillen et al., 2002) and glycolytic oscillators (Wolf and Heinrich, 1997; Wolf et al., 2000). The coupling of these networks is realized by substances which diffuse through the common extracellular medium. These substances or complexes in turn affect each individual network after mixing and diffusing in the common medium. In contrast to the direct coupling, such as a neural network where each neuron directly connects with others, such indirect coupling provides a strategy to regulate the dynamics of the individual oscillators just by artificially controlling the coupling substances, rather than directly controlling each individual oscillator. Generally, noises are everywhere in a biochemical system because of the intrinsically or extrinsically stochastic nature of the reactions involved (Thattai and van Oudenaarden, 2001). A common source of noise can also induce synchronization (Chen et al., 2005; Teramae and Tanaka, 2004; Zhou et al., 2005). However, complete understanding of its origin and how to exploit it to regulate cellular functions require further investigation. Therefore, a more realistic artificial control strategy which can be easily implemented in experiments and further clinic applications to control pathological rhythms are increasingly demanded.
The main purpose of this paper is to construct an impulsive control system (Yang, 2001) to model the process of periodically injecting the coupling substances, which are shared by all genetic oscillators, with constant or random impulsive amounts into the common extracellular medium, and further study its effects of such control scheme on the dynamics of individual oscillators. Specifically, we study the impulsive control scheme for indirectly coupled genetic networks, and derive the threshold of synchronization induced by the periodic input. We can synchronize a multicellular network to a specific collective behavior, by changing the frequency and amplitude of the periodic stimuli. The multicellular network of coupled repressilators is used to show the effectiveness of the proposed method. The results provide not only a new prospective to understand the interactions between the external stimuli and intrinsic physiological rhythms but also may lead to development of realistic artificial control strategy or even medical therapy. For example, when there is a shift of light–dark cycles generated by visiting a different time zone, the cycles can still entrain the intrinsic rhythms (Daan et al., 1984; Jewett and Kronauer, 1998). In addition, there are a number of ways by which insight and results from the analysis of the entrained synchronization can be transferred to medical use. Because bodily functions show distinct periodicity, temporal schedules for drug administration might be optimized. In cancer chemotherapy, treatments could be based on the circadian rhythm of cell division (Mormont and Lévi, 1997).
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2 THE MODEL |
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TOPABSTRACT1 INTRODUCTION2 THE MODEL3 RESULTS4 CONCLUSIONREFERENCES |
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The repressilator is a network of three genes, the products of which inhibit the transcription of each other in a cyclic way (Elowitz and Leibler, 2000). Its sufficient conditions for periodic oscillations with time delays are given in (Wang et al., 2005). Garcia-Ojalvo et al. (2004) proposed a modular addition to the repressilator with the aim of coupling a population of cells containing this network, and analyzed the self-synchronization numerically. The collective rhythms of the multicellular network were further studied, based on the Lyapunov function method (Wang and Chen, 2005). In this paper, we adopt the coupled repressilators as an example. The basic mechanism of communication among cells is based on the quorum-sensing, which was first discovered in the bacterium Vibrio fischeri, a bioluminescent organism that lives in symbiosis with certain marine hosts forming part of specialized light organs. These bacteria exhibit collective rhythms mainly through two proteins. The first one, LuxI, synthesizes a small molecule known as an autoinducer (AI), which diffuses freely through the cell membrane. The second protein, LuxR, binds to the AI molecule to form a complex, which in turn activates transcription of various genes. Recent studies indicated that the quorum sensing can actually be engineered and utilized as an intercellular signaling system in Escherichia coli (Kobayashi et al., 2004; You et al., 2004).
The mRNA dynamics in cell i 
{1,2, ... , N} are governed by repressible transcription for all three genes of the repressilator plus transcription activation of the additional copy of the lacI and their degradation:
 | (1) |
is the dimensionless transcription rate in the absence of repressor and k is the maximal contribution to lacI transcription in the presence of the saturating amount of AI. n is the Hill coefficient.
The dynamics of proteins TetR, CI and LacI in each cell are given by
 | (2) |
 | (3) |
measures the diffusion rate of AI across the cell membrane. Note that, by assuming equal lifetimes of the TetR and LuxI, we use the same variable Ai of TetR to represent LuxI in Equation (3) due to their identical dynamics (Garcia-Ojalvo et al., 2004). As in references (Garcia-Ojalvo et al., 2004; McMillen et al., 2002), our study is also based on the assumption that the release of AI is fast enough with respect to the timescales of the other components so that AI quickly becomes homogeneous to establish an average level or a mean field 
in the common environment. For the case of slow diffusion or inhomogeneity, diffusion-reaction equations or other techniques are more appropriate to describe the spatially heterogeneous culture. To handle inhomogeneity in a more accurate manner is important on studying cellular dynamics, and we leave this for our future work.
The coupling depends on the synchronizing factor in the extracellular medium. Let Se represent the extracellular concentration of AI in the common environment. Its dynamics is governed by
 | (4) |
e is the diffusion rate and de gives the rate of decay of the AI in the extracellular environment, which is assumed to be homogeneous. Therefore, the multicellular network is described by Equations (1)–(4). Clearly, the system is different from that of so-called star-type coupling, and also from that of a neural network. The main differences are in the description of interaction among cells, e.g. any two cells are not directly coupled but indirectly interacted through Se, i.e. a coupling substance in the extracellular medium, which is shared by all cells. Such coupling is more biologically plausible in many biological systems, and has been studied by many researchers (Chen et al., 2005; Garcia-Ojalvo et al., 2004; McMillen et al., 2002; Wolf and Heinrich, 1997; Wolf et al., 2000; You et al., 2004). The diffusion of extracellular AI molecules into the cells provides a mechanism of intercellular coupling. When the cell density is large enough, perfect locking and collective rhythms can be observed (Garcia-Ojalvo et al., 2004). Since the rhythms interact with each other among fluctuating external environment, disruption of the rhythmic processes beyond normal bounds, emergence of abnormal rhythms, or large fluctuations of the external environment are often associated with loss of the collective rhythms. Moreover, diseases can also lead to alternations from normal collective rhythms to abnormal non-collective ones. Therefore, an artificial control strategy is demanded to compensate the inefficiency of the coupling, control the pathological non-collective rhythms and even treat human diseases.
Based on the impulsive control theory (Yang, 2001), an artificial control strategy is proposed in this paper. When spontaneous synchronization cannot be achieved due to inefficiency of the coupling, an external input can be used to compensate the inefficiency and thus induce controlled synchronization. Therefore, the external input can be viewed as an amplifier of the coupling. Now we model this process according to Equations (1)–(4) by introducing a periodic input which injects the chemical substance AI into the common extracellular medium at fixed instants. That is, instead of Equations (1)–(4), we consider the following impulsive control system
 | (5) |
Se(t) = Se(t+) – Se(t) = 
is the impulsive input or injection amount of AI into the common extracellular medium at control instants t = n
; 
; is the period of the impulsive effect. Clearly, Se(t) gives a sudden change of AI in the extracellular medium at instants t = n. For the impulsive control system of Equations (5), its dynamics are determined by the the original system, i.e. Equations (1)–(4), injection amount , and injection period . We use Equations (5) to model the process of periodical input which injects the chemical substance AI into the common extracellular medium, and study the effects of external input on the dynamics of individual cells. Generally, the external input can profoundly affect the dynamics of AI in the medium, thus also affect the dynamics of individual cells due to the indirect coupling. The scheme of the impulsive control system governed by Equations (5) is shown in Figure 1.
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3 RESULTS |
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TOPABSTRACT1 INTRODUCTION2 THE MODEL3 RESULTS4 CONCLUSIONREFERENCES |
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3.1 Synchronizing a population of noisy repressilators
In the absence of external input, i.e.
= 0, when 
0, Equations (5) consist of a population of uncoupled limit-cycle oscillators. The population of oscillators contain substantial differences from cell to cell because of the intrinsically stochastic nature of reactions involved, giving rise to a relatively broad distribution in the periods of the individual oscillators. We consider a system of 10 000 cells, where ß follows a Gaussian distribution with a mean of ß = 2 and SD of 5%, i.e. ß = 0.1. Most periods range from 8.5 to 11.5 min, as shown in Figure 2a. The temporal evolution of the protein TetR concentration in 10 of those cells is plotted in Figure 2b, showing that collective rhythms cannot be observed under these conditions. In other words, spontaneous synchronization cannot be achieved due to the coupling inefficiency.
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and Different from the method used in (Garcia-Ojalvo et al., 2004), where efficient self-synchronization can occur by increasing the cell density, we try to investigate how to compensate the coupling inefficiency by an artificial control strategy. Figure 2d shows that the external input can indeed entrain the intrinsic rhythms or induce collective rhythms although their natural periods are broadly distributed. The impulse period is
= 10 min, which is close to the mean period of all oscillators. Thus, a transition from desynchronization to synchronization exists for an appropriate frequency and amplitude of the periodic external input. The synchronized oscillators lead to a single period, which is identical for all oscillators and determined by the external input cycle, as shown in Figure 2c. Because not all of the oscillators have the same individual period, a perfect synchronization cannot be achieved and phase differences between some oscillators still persist, as shown in Figure 2d. The system behaves as a macroscopic clock with a well-defined period set by the external input cycle, although it is composed of a widely varied collection of oscillators.
The dynamics of the impulsive control system may strongly depend on the frequency and amount of the external input. Therefore, it is important to evaluate the threshold of synchronization by the impulsive control strategy. The synchronization regions as functions of and for different coupling coefficients are shown in Figure 3. The system exhibits synchronization in a wide region of the parameter space, as shown in Figure 3a. However, for a relatively smaller coupling, a larger impulsive amount is required to synchronize the oscillators for the specific impulsive periods, as shown in Figure 3b. Note that the impulsive control system can be easily synchronized when the impulsive period is close to the mean period of the oscillators. Furthermore, the more the impulsive amount of the external input is, the larger the range of the impulsive period for synchronization can be used. In the presence of an appropriate impulsive control, when the impulsive period is close to the mean period of all oscillators, the characteristic oscillations of the controlled synchronization do not change qualitatively except the dynamics of AI in the intracellular and extracellular medium, which is changed drastically by the external input, as shown in Figure 2b and d.
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A comparison of the synchronization regions shows that for the same impulsive period, the minimum impulsive amount required to synchronize a population of oscillators decreases with the increase of the coupling strength, which means that for a relatively larger coupling strength with a specific impulsive period, a smaller impulsive amount is needed to compensate the coupling inefficiency. Therefore, although the coupling itself is insufficient to induce spontaneous synchronization, it still plays an important role in the synchronization. In other words, the synchronization is induced by the coupling together with the external input, rather than the external input itself. When spontaneous synchronization cannot occur due to weak coupling or other reasons, external input as an artificial control strategy can be adopted.
The impulsive control strategy has two beneficial effects. One is that it can compensate the coupling inefficiency when spontaneous synchronization cannot be achieved, i.e. the external input can be used as a coupling amplifier for controlled synchronization. Moreover, the amount and period of the external input are independent of the state variables, therefore, when using the periodic input to increase its effectiveness at control, we need not measure the state variables at the control instants, which makes the method biologically plausible and easy to implement in experiments and further medical use. The other is that it can reduce the noisiness of the multicellular network, effectively transforming an ensemble of noisy clocks into a very reliable collective oscillator. Our findings demonstrate an efficient way to synchronize multicellular networks by an artificial control strategy and also provide a powerful mechanism for noise resistance.
Note that the model we study here is a coupled system. Although mathematically we can use the same external input to directly entrain each individual oscillator independently, it is neither biologically plausible nor can be easily implemented. The coupling substance in the extracellular environment which is shared by all individual oscillators, and the coupling inefficiency between the common environment and the individual oscillators are two important factors of our model. Owing such properties, the dynamics of each individual oscillator can be indirectly controlled by just perturbing the coupling substance in the common environment through a periodic input, which can be more easily implemented in contrast to direct control on each cell. In such a simple control manner, the coupling inefficiency can be compensated and eventually the synchronization can be derived.
3.2 Impulse-induced dynamics
Besides the synchronization of the multicellular network, the impulsive control can also initiate different dynamics, depending on the frequency and amplitude of the external input. The impulsive control theory has been used to completely stabilize chaotic systems by a linear but state variable-dependent control law, i.e. the control law is also zero at the equilibria (Yang, 2001). Although the state-independent control law is easy to implement in experiments, it is difficult to analyze its stabilizing effect because equilibria do not exist. However, we still find that the constant impulsive control has the similar effect. Moreover, the external input may also induce complex dynamics.
To examine effects of the constant impulsive input on the dynamics, we plot time evolution of 10 noisy cells in Figure 4a. The coupling AI in the extracellular space, i.e. Se, oscillates in a stable cycle with period = 0.5 min, which is determined by the external input cycle. In contrast, the oscillations of other components are suppressed. In other words, the periodic input of the AI into the extracellular medium brings the oscillators out of their oscillatory domains. Note that because the control law, i.e. Se = 
is independent of state variables, there is no trivial solution for the impulsive control system. Thus, different external input may bring the system to different states. Moreover, although there is a noticeable irregularity in the oscillatory behavior of each individual cell, i.e. the periods and amplitudes of individual cells are broadly distributed, a population of cells can still be brought to an approximately identical state, as shown in Figure 4a.
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i 
0.01, where Because many bodily functions show a distinct periodicity, schedules for drug administration might be optimized. Intriguing strategies to minimize the effects of jet lag have been developed, based on experimental studies of resetting the circadian oscillators by modifying the exposure to light after travel (Winfree, 1980). The results suggest that appropriate external input can suppress the oscillatory behaviors. In other words, there is a notable transition from different oscillatory behaviors to an identical non-oscillatory state, which provides a new way to reset a population of different oscillators to an identical state.
The suppression of oscillations may depend on the amount and period of the external input. The maximum , denoted as max, as a function of is shown in Figure 4b. For a given , when < max, the oscillations of all components except Se can be suppressed. As shown in Figure 4b, when we increase , max will increase first and then decrease at about = 3.2. When compared with Figure 3, Figure 4b shows that relatively smaller and period of the external input can bring the impulsive control system to non-oscillatory states.
If we continue to increase , the external input may induce a variety of complex dynamical behaviors, such as chaos. Because it is not straightforward to distinguish deterministic chaotic dynamics from noisy dynamics, we use 10 identical cells to eliminate the effects of noise, although it always exists in practice. The chaotic synchronization induced by the external input at = 3 min and = 2 is shown in Figure 5.
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3.3 Periodic input with random impulsive amounts
Till now, we focus mainly on the periodic external input with a constant impulsive amount. Because the rhythms arise from a stochastic, biological mechanism interacting with a fluctuating environment, the strictly constant impulsive amount is not plausible. Therefore, it is important to consider the effect of the periodic input with random impulsive amounts on the dynamics of a population of noisy oscillators. In the absence of stochastic input, the periods and amplitudes of the oscillations are heterogeneous and collective dynamics do not occur. However, under conditions of stochastic input, i.e. periodic input with random variation of the substance input, we show that collective dynamics may occur, depending on the stochastic input. The input
is randomly chosen from the interval [a, b]. Two situations are shown in Figure 6, which suggests that a population of noisy oscillators can be entrained by a periodic source of the coupling substance and yield sustained oscillations with irregular waveform and a stable period, which is determined by the relative magnitude of the period of the external input. In contrast to the stability of the period, both the irregularity of the oscillators induced by the intrinsic or extrinsic noise and the random impulsive amounts render the amplitude irregular.
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3.4 Two-stage synchronization for identical oscillators
Types and ranges of entrainment of a population of noisy oscillators by a periodic substance input have been studied. Appropriate external input may yield regular or irregular oscillations with stable periods determined by the magnitude of the period of the input, and even complete suppression of oscillations. Therefore, the external input can never be stopped to keep the entrainment of a population of noisy oscillators due to the stochastic nature. Moreover, the period of the controlled synchronization is determined by that of the input.
However, the situation is totally different for a population of identical oscillators. Whatever the entrained dynamics are, when the trajectories of the oscillators are sufficiently close, the external injection can be terminated. The nature of a periodic oscillation in each individual oscillator keeps the difference between the trajectories of individual oscillators so that it will not diverge, which is different from sensitive dependence on initial conditions for chaotic oscillators. Based on such an analysis, we propose a two-stage synchronization scheme, by which the initial conditions of individual oscillators can be reset and the final synchronization state is independent of the external input, although they cannot be synchronized in the absence of external input.
The two-stage synchronization scheme is described as follows:
- Use the periodic external input with constant or random impulsive amounts to reduce the differences of trajectories between individual oscillators until they approach as close as desired. The individual oscillators may have very different initial conditions. We can compute a bound for the time required.
- After all trajectories are close enough, we stop the external input so that all oscillators can be synchronized to an appropriate oscillation determined by the isolated oscillator in the absence of external input.
The key problem in the two-stage synchronization is to ensure that the trajectories of individual oscillators are as close as desired in stages 1–2. With the weak coupling, each individual oscillator is still periodically oscillatory, therefore, when starting from nearby initial conditions, the difference between different trajectories will not diverge. This means that in the second stage, the synchronization errors of the indirectly coupled multicellular network can be held when the error at which we switch from the first stage is sufficiently small. To obtain the time instant to stop the external input, we need to determine the tolerable error that can ensure that all trajectories are close enough when switching to stage 2.
We now show how to apply the two-stage synchronization approach for the multicellular network of the indirectly coupled repressilators. First, we use the external input to decrease the differences of individual trajectories. The network can be entrained by a periodic source of substance input with random impulsive amount . Each individual oscillator has a different initial condition. When t is close to 100 min, the trajectories are close enough, and the input can be stopped. Two cases are shown in Figure 7 for small and large random impulsive amount , respectively. The impulsive control is stopped at t = 100 min and perfect synchronization can be quickly obtained.
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4 CONCLUSION |
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TOPABSTRACT1 INTRODUCTION2 THE MODEL3 RESULTS4 CONCLUSIONREFERENCES |
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From an engineering perspective, the control of cellular functions through artificial control strategy is an intriguing possibility. In this paper, we have shown how external input can be used to control the dynamics of individual oscillators in a systematic way. Although the main focus of this work is on a multicellular network of coupled repressilators, our approach is generally applicable to other networks or other coupling schemes, such as star coupling.
An important element of our control scheme is the coupling substance, which is shared by individual oscillators. Therefore, the dynamics of individual oscillators can be indirectly controlled by just controlling the coupling substance through a periodic input with constant or random impulsive amounts. Our results suggest that the impulsive control strategy can be used to entrain multicellular networks and initiate collective rhythms, even with the broadly distributed periods and amplitudes of individual oscillators. The impulsive control mechanism indicates that collective rhythms can be easily obtained by the external input when the impulsive period is close to the mean period of a population of noisy oscillators, otherwise, some other dynamics such as suppression of oscillations or even chaos may occur. A two-stage synchronization scheme is proposed based on the assumption that all oscillators are identical. Hence, when all trajectories are close enough, the external input can be stopped. However, owing to stochastic nature of the reactions involved, substantial variability and noticeable irregularity may exist. Therefore, to entrain a population of noisy oscillators, a pulse-generating network (Basu et al., 2004) is needed to generate ceaseless stimuli with an appropriate frequency and amplitude, which should be an important future problem.
There are a number of potential applications for our approach. For the effective treatment of many diseases, which often cause alternations from normal to pathological rhythms or from collective rhythms to non-collective ones, many cells need to be regulated in some systematic manner. Thus the development of artificial control strategy could have significant clinical or experimental implications. Moreover, better understanding of the interactions between an external periodic input and intrinsic physiological rhythms may lead to the development of better medical therapy.
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Acknowledgments |
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This research is partially supported by Grant-in-Aid for Scientific Research on Priority Areas 17022012 from MEXT of Japan.
Conflict of Interest: none declared.
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FOOTNOTES |
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Associate Editor: Satoru Miyano
Received on April 13, 2006; accepted on May 5, 2006
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