Entrained Collective Rhythms of Multicellular Systems: Partial Impulsive Control Strategy

Abstract

This paper is concerned with the study of entrained collective rhythms of multicellular systems by using partial impulsive control strategy. The objective is to design an impulsive controller based on only those partially available cell states, so that the entrained collective rhythms are guaranteed for the multicellular systems with cell-to-cell communication mechanism. By using the newly developed impulsive integrodifferential inequality, the sufficient conditions are derived to achieve the entrained collective rhythms of multicellular systems. A synthetic multicellular system with simulation results is finally given to illustrate the usefulness of the developed results.

1. Introduction

Complex physiological rhythms are ubiquitous in living organisms, which are central to life, such as our daily cycle of waking and sleeping and the beating of our hearts. Collective rhythms are normally generated by thousands of divers clock cells which manage to function in a coherent oscillatory state [1, 2]. In fields ranging from circadian biology to endocrinology, however, it remains an exciting challenge to understand how collective rhythms emerge in multicellular structures [3–7].

Elucidating the collective dynamics of multicellular systems not only is essential for the understanding of the rhythmic phenomena of living organisms at both molecular and cellular levels but also has many potential applications in bioengineering areas. For example, in cancer chemotherapy, treatments could be based on the circadian rhythm of cell division [8]. Over the past decade or so, many researchers have paid a great deal of attention to study the collective dynamics of multicellular systems. For instance, in [9–12], the authors considered stability of genetic networks and neural networks. In [3], the authors pointed out that intercell signaling mechanism does lead to synchronous behavior across a population of cells. In [13], after making real-time analysis of the gene expression, the authors showed the synchronized rhythms of clock gene transcription across hundreds of neurons within the mammalian suprachiasmatic nucleus (SCN) in organotypic slice culture. In addition, based on the Lyapunov stability theory, the collective rhythms of multicellular systems were further studied in [14]. For the other relevant results, please see [15–17].

Although there are significant advances on elucidating the collective behaviors of biological organisms in recent years, the essential mechanisms from which the collective rhythms arise remain to be fully understood. It is well known that coupling among cells is not sufficient to achieve collective rhythms. In fact, the collective rhythms of multicellular systems are far away from being well understood and warrant further and insightful study.

On the other hand, experimental results have already shown that external stimuli play an important role in achieving the collective rhythms. In [18], physiological rhythms were induced by regular or periodic inputs occurring in the context of medical devices. In [19], an external voltage was applied to enhance the synchronization of electronic synthetic genetic networks. In [20], it was shown that a specific collective behavior could be realized by changing the frequency and amplitude of the periodic stimuli. Another well-known example is that organisms usually display a circadian rhythm, where the key processes show a 24-hour periodicity entrained to the light-dark cycle [21]. In [22], the authors studied the rhythmic process of the circadian oscillators under the effect of the daily light-dark cycle. Furthermore, from the view of impulsive control systems, collective behaviors of coupled systems were investigated and some interesting results have been obtained in [23–27], and, for the other relevant results, please refer to [28–32] and the references therein.

However, in the above-mentioned results, one basic assumption is that the external stimuli are applied to all the cells in the community, that is very expensive or unrealistic in practice. Actually, in many practical medical cases, only partial specific cells could be detected and utilized. In these situations, the external stimuli are applied to only those cells in the community. To the best of our knowledge, there are few results in the open literature on the entrained collective rhythms of multicellular systems by applying impulsive control based on the partially available cell states.

This paper is to study the entrained collective rhythms of multicellular systems with only partially available cell states. By using the newly developed impulsive integrodifferential inequality, a new criterion is derived to ensure the entrained collective rhythms of multicellular systems. It is shown that when the spontaneous synchrony cannot be achieved, an appropriate periodic stimulus could achieve a collective rhythm even only with partially available cell states. It is noted that the proposed partial impulsive control method can be also easily extended to study other complex systems.

The rest of the paper is organized as follows. Section 2 formulates the problem of the entrained collective rhythms and provides some useful lemmas. Section 3 presents the main results for entrained collective rhythms of multicellular systems. A synthetic multicellular system will be employed to illustrate the effectiveness of the developed results in Section 4, which is followed by conclusions in Section 5.

2. Model Description and Problem Formulation

To make it easy for the readers, let us start from a single cell model of the form 𝑥̇(𝑡)=−𝐴𝑥(𝑡)+𝑓(𝑥(𝑡)), (1) where 𝑥(𝑡)∈ℝ^𝑛+ represents the concentrations of proteins, RNAs, and other chemical complexes, 𝐴 is the positive diagonal matrix denoting the degradation and dilution rate, and 𝑓(𝑥(𝑡)) is the complex regulatory function, which usually is of the Michaelis-Menten or Hill form.

Remark 1. It is known that many biological models can be represented by (1), such as the Goodwin model [33] and the toggle switch [34]. Furthermore, the regulatory function 𝑓(𝑥) in model (1) is usually monotonically increasing or decreasing.

Without loss of generality, the regulatory function is always assumed to satisfy the following assumption.

Assumption 2. The regulatory function 𝑓(𝑥) in (1) satisfies 𝑓_𝑖(𝑎)−𝑓_𝑖(𝑏)𝑓_𝑖(𝑎)−𝑓_𝑖(𝑏)−𝑙_𝑖(𝑎−𝑏)≤0, (2) for all 𝑎,𝑏∈𝑅,𝑎≠𝑏, and 𝑙_𝑖>0.
Consider multicellular systems with cell-to-cell communication mechanism described as follows: 𝑥̇_𝑖(𝑡)=−𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑖=1,…,𝑁, (3) where 𝑥_𝑖(𝑡)∈ℝ^𝑛+ is the state of the 𝑖th cell, denoting the concentrations of chemical complexes in this cell, and 𝑁 is the total cell number of the entire community. The third term in model (3) describes the capability of cells to communicate with each other in order to coordinate the behavior of the entire community. 𝐺=(𝐺_𝑖𝑗)_𝑁×𝑁 is the coupling structure matrix that represents the communications between different cells, and Γ is the inner coupling structure that represents the connections of different chemical complexes in one cell. 𝐺 satisfies the diffusive coupling condition 𝐺_𝑖𝑗𝐺≥0,for𝑖≠𝑗,_𝑖𝑖=−^𝑁_{𝑗=1,𝑗≠𝑖}𝐺_𝑖𝑗,𝑖=1,2,…,𝑁. (4) It can be noted that such coupling is biologically plausible in many biological systems, such as the quorum sensing mechanism in bacteria [2, 35].
Suppose only 𝑙 cell states in the community are measurable for the multicellular systems (3). Consider the following linear impulsive controller based on those 𝑙 measurable cell states: 𝑢_𝑖=^∞_𝑘=1𝐸_𝑖𝑘𝑥_𝑖−𝑥𝛿𝑡−𝑡_𝑘,𝑖=1,2,…,𝑙,𝑘∈ℕ, (5) where 𝑥 is the state of the isolated cell described in (1), 𝑙 is the number of the measurable cell states, 𝐸_𝑖𝑘 is the gain matrix, and 𝛿(𝑡−𝑡_𝑘) is the Dirac impulse function with discontinuity points 𝑡₁<𝑡₂<⋯<𝑡_𝑘<⋯,lim_𝑘→∞𝑡_𝑘=∞.
Then the impulsive-controlled multicellular systems with partial states can be described by the following impulsive differential equation: 𝑥̇_𝑖(𝑡)=−𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑡∈𝑡_𝑘−1,𝑡_𝑘,Δ𝑥_𝑖𝑡_𝑘=𝑥_𝑖𝑡⁺_𝑘−𝑥_𝑖𝑡⁻_𝑘=𝐸_𝑖𝑘𝑥_𝑖−𝑥,𝑘=1,2,…,𝑡=𝑡_𝑘,𝑥̇𝑖=1,…,𝑙,_𝑖(𝑡)=−𝐴𝑥_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡)+𝑐^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑥_𝑗(𝑡),𝑖=𝑙+1,…,𝑁. (6)
Defining 𝑒_𝑖(𝑡)=𝑥_𝑖(𝑡)−𝑥(𝑡), one can obtain the following error system: 𝑒_𝑖̇(𝑡)=−𝐴𝑒_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡)−𝑓(𝑥(𝑡))+𝑐^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑒_𝑗(𝑡),𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑒_𝑖𝑡⁺_𝑘=𝐼+𝐸_𝑖𝑘𝑒_𝑖𝑡_𝑘,𝑘=1,2,…,𝑡=𝑡_𝑘𝑒̇,𝑖=1,…,𝑙,_𝑖(𝑡)=−𝐴𝑒_𝑖(𝑡)+𝑓𝑥_𝑖(𝑡)−𝑓(𝑥(𝑡))+𝑐^𝑁_𝑗=1𝐺_𝑖𝑗Γ𝑒_𝑗(𝑡),𝑖=𝑙+1,…,𝑁. (7)

Then the problem of entrained collective rhythms is to design the partial impulsive controller (5) such that the stability of the error system (7) is guaranteed. Before presenting the main results, some useful lemmas are introduced in advance.

Lemma 3 (see [36]). If 𝑃∈ℝ^𝑛×𝑛 is a positive definite matrix and 𝑄∈ℝ^𝑛×𝑛 is a symmetric matrix, then 𝜆_min𝑃⁻¹𝑄𝑥^T𝑃𝑥≤𝑥^T𝑄𝑥≤𝜆_max𝑃⁻¹𝑄𝑥^T𝑃𝑥,∀𝑥∈ℝ^𝑛, (8) where 𝜆_min(⋅) and 𝜆_max(⋅) are the minimum and maximum eigenvalues of the matrix, respectively.

Lemma 4. For positive scalars 𝛿>0,  𝜇>0,  and  𝜌>0,  if  𝑢(𝑡)  satisfies𝑢̇(𝑡)≤ℎ(𝑡,𝑢(𝑡))+𝛿^𝑡₀𝑒^{−𝜌(𝑡−𝑠)}𝑢(𝑠)𝑑𝑠,𝑡≠𝑡_𝑘,𝑢𝑡⁺_𝑘≤𝜇𝑢𝑡_𝑘,𝑡=𝑡_𝑘,𝑢(0)≤𝑢₀, (9) where ℎ(𝑡,𝑢(𝑡)) is a continuous function, then one has 𝑢(𝑡)≤𝑢_𝜖(𝑡),𝑡≥0 for 𝜖≥0, where 𝑢_𝜖(𝑡) is the solution to the following impulsive integrodifferential equation: 𝑢̇_𝜖(𝑡)=ℎ𝑡,𝑢_𝜖(𝑡)+𝛿^𝑡₀𝑒^{−𝜌(𝑡−𝑠)}𝑢_𝜖(𝑠)𝑑𝑠+𝜖,𝑡≠𝑡_𝑘,𝑢_𝜖𝑡⁺_𝑘=𝜇𝑢_𝜖𝑡_𝑘,𝑡=𝑡_𝑘,𝑢_𝜖(0)=𝑢₀. (10)

Proof. Firstly, we prove 𝑢(𝑡)≤𝑢_𝜖(𝑡),𝑡∈0,𝑡₁. (11) If argument (11) is not right, then there exists 𝑡̃∈(0,𝑡₁) such that 𝑢𝑡̃>𝑢_𝜖𝑡̃. (12) Considering the continuity of 𝑢(𝑡), 𝑢_𝜖(𝑡) on (0,𝑡₁], there must exist 𝑡^∗∈(0,𝑡)̃ such that 𝑢𝑡^∗=𝑢_𝜖𝑡^∗,𝑢̇𝑡^∗≥𝑢̇_𝜖𝑡^∗,𝑢(𝑡)≤𝑢_𝜖(𝑡),∀𝑡≤𝑡^∗; (13) then it yields 𝑢̇𝑡^∗≤ℎ𝑡^∗,𝑢𝑡^∗+𝛿^𝑡∗₀𝑒^{−𝜌(𝑡∗−𝑠)}𝑢(𝑠)𝑡𝑑𝑠<ℎ^∗,𝑢_𝜖𝑡^∗+𝛿^𝑡∗₀𝑒^{−𝜌(𝑡∗−𝑠)}𝑢_𝜖(𝑠)=𝑢̇𝑑𝑠+𝜖_𝜖𝑡^∗, (14) which contradicts the condition 𝑢̇(𝑡^∗)≥𝑢̇_𝜖(𝑡^∗), so (11) holds.
Suppose 𝑢(𝑡)≤𝑢_𝜖(𝑡), for all 𝑡∈(0,𝑡_𝑘]; then by 𝑢(𝑡⁺_𝑘)=𝜇𝑢(𝑡_𝑘)≤𝜇𝑢_𝜖(𝑡_𝑘)=𝑢_𝜖(𝑡⁺_𝑘), similarly, one has 𝑢(𝑡)≤𝑢_𝜖(𝑡), for all 𝑡∈(𝑡_𝑘,𝑡_𝑘+1]. By using the mathematical induction method, one can conclude 𝑢(𝑡)≤𝑢_𝜖(𝑡), for all 𝑡∈(0,𝑡_𝑘] for any positive integer 𝑘. The proof is thus complete.

Lemma 5 (Grownwall-Bellman Inequality [37]). Let 𝛼(𝑡) be a real value continuous function and 𝛽(𝑡) a nonnegative continuous function on [𝑎,𝑏]. If a continuous function 𝑢(𝑡) has the property that 𝑢(𝑡)≤𝛼(𝑡)+^𝑡_𝑎𝛽(𝑠)𝑢(𝑠)𝑑𝑠,𝑎≤𝑡≤𝑏, (15) then on [𝑎,𝑏] one has 𝑢(𝑡)≤𝛼(𝑡)+^𝑡_𝑎𝛼(𝑠)𝛽(𝑠)exp^𝑡_𝑠𝛽(𝑟)𝑑𝑟𝑑𝑠. (16)

Definition 6. The multicellular system (6) is said to achieve collective rhythms with the designed partial impulsive controller, if there exist scalars 𝜌>0 and 𝑀>0 such that lim_𝑡→∞‖𝑋(𝑡)‖²≤𝑀𝑒^−𝜌𝑡‖𝑋(0)‖², (17) where 𝑋(𝑡)=[𝑥^T₁−𝑥^T,…,𝑥^T_𝑁−𝑥^T]^T and 𝑋(0) is the initial condition.

3. Main Results

In this section, by using the proposed impulsive integrodifferential inequality, a sufficient condition guaranteeing the entrained collective rhythms of multicellular systems is derived.

Theorem 7. For a given scalar 𝜇∈(0,1), if there exist matrices 𝑃_𝑖>0, 𝑖=1,2,…,𝑙,𝑄_𝑖>0, 𝑖=𝑙+1,…,𝑁, scalars 𝜆_𝑖>0, 𝜇_𝑖>0, 𝑖=1,2,…,𝑙,𝜂_𝑖>0, 𝑖=𝑙+1,…,𝑁, and positive scalars 𝜖₁, 𝜖₂, 𝛿₁, and 𝛿₂ such that −𝐴^T𝑃_𝑖−𝑃_𝑖𝐴+𝜆_max𝑃_𝑖𝜖₁𝐼+𝜖₁⁻¹𝐿^T𝐿+𝑐𝛿₁^𝑁_𝑗=1𝐺_𝑖𝑗+𝑐𝛿₁^𝑙−1_𝑗=1𝜆_max𝑃_𝑗𝐺_𝑗𝑖Γ^TΓ≤𝜆_𝑖𝑃_𝑖,𝐼+𝐸_𝑖𝑘^T𝑃_𝑖𝐼+𝐸_𝑖𝑘≤𝜇_𝑖𝑃_𝑖, (18)(19) for 𝑖=1,…,𝑙, and −𝐴^T𝑄_𝑖−𝑄_𝑖𝐴+𝜆_max𝑄_𝑖𝜖₂𝐼+𝜖₂⁻¹𝐿^T𝐿+𝑐𝛿₂^𝑁_𝑗=1𝐺_𝑖𝑗+𝑐𝛿₂^𝑁−1_𝑗=𝑙+1𝜆_max𝑄_𝑗𝐺_𝑗𝑖Γ^TΓ≤−𝜂_𝑖𝑄_𝑖, (20) for 𝑖=𝑙+1,…,𝑁, and, for any impulsive time sequence {𝑡_𝑘} satisfying 𝛿∶=sup_𝑘𝑡_𝑘+1−𝑡_𝑘<ln1/𝜇/,𝜂+𝜆+𝛽𝛾𝜂𝜇 (21) for 𝑘=0,1,2,…, where 𝜇=max(𝜇_𝑖), 𝜆=max(𝜆_𝑖), 𝑖=1,…,𝑙, and 𝜂=min(𝜂_𝑖), 𝑖=𝑙+1,…,𝑁, then the entrained collective rhythms of multicellular systems (6) are achieved.

Proof. Consider the following Lyapunov function: 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡), (22) where 𝑉₁(𝑡)∶=^𝑙_𝑖=1𝑒^T_𝑖(𝑡)𝑃_𝑖𝑒_𝑖(𝑡),𝑉₂(𝑡)∶=^𝑁_𝑖=𝑙+1𝑒^T_𝑖(𝑡)𝑄_𝑖𝑒_𝑖(𝑡), (23) where 𝑃_𝑖,𝑖=1,2,…,𝑙 and 𝑄_𝑖,𝑖=𝑙+1,…,𝑁 are positive definite matrices to be determined.
For any 𝑡∈(𝑡_𝑘,𝑡_𝑘+1], 𝑘∈ℕ, taking the Dini derivative along the trajectories of (7), we have 𝐷⁺𝑉₁(𝑡)=^𝑙_𝑖=1𝑒^T_𝑖̇(𝑡)𝑃_𝑖𝑒_𝑖(𝑡)+𝑒^T_𝑖(𝑡)𝑃_𝑖𝑒_𝑖̇(𝑡)=^𝑙_𝑖=1𝑒^T_𝑖(𝑡)−𝐴^T𝑃_𝑖−𝑃_𝑖𝐴𝑒_𝑖(𝑡)+2𝑒^T_𝑖(𝑡)𝑃_𝑖𝑓𝑥_𝑖(𝑡)−𝑓(𝑥(𝑡))+2𝑐^𝑁_𝑗=1𝑒^T_𝑖(𝑡)𝑃_𝑖𝐺_𝑖𝑗Γ𝑒_𝑗(𝑡). (24) It follows from Assumption 2 that 2𝑒^T_𝑖(𝑡)𝑃_𝑖𝑓𝑥_𝑖(𝑡)−𝑓(𝑥(𝑡))≤𝜆_max𝑃_𝑖𝜖₁𝑒^T_𝑖(𝑡)𝑒_𝑖(𝑡)+𝜖₁⁻¹𝑒^T_𝑖(𝑡)𝐿^T𝐿𝑒_𝑖(𝑡),2𝑐^𝑁_𝑗=1𝑒^T_𝑖(𝑡)𝑃_𝑖𝐺_𝑖𝑗Γ𝑒_𝑗(𝑡)≤𝑐𝜆_max𝑃_𝑖^𝑁_𝑗=1𝐺_𝑖𝑗×𝛿₁𝑒^T_𝑖(𝑡)𝑒_𝑖(𝑡)+𝛿₁⁻¹𝑒^T_𝑗(𝑡)Γ^TΓ𝑒_𝑗(𝑡), (25) where 𝜖₁>0, 𝛿₁>0 and 𝐿=diag(𝑙₁,…,𝑙_𝑛). One also has 𝑐𝛿₁^𝑙−1𝑁_𝑖=1𝑗=1𝜆_max𝑃_𝑖𝐺_𝑖𝑗𝑒^T_𝑗(𝑡)Γ^TΓ𝑒_𝑗(𝑡)=𝑐𝛿₁^𝑁−1𝑙_𝑖=1𝑗=1𝜆_max𝑃_𝑗𝐺_𝑗𝑖𝑒^T_𝑖(𝑡)Γ^TΓ𝑒_𝑖(𝑡)=𝑐𝛿₁^𝑙−1𝑙_𝑖=1𝑗=1𝜆_max𝑃_𝑗𝐺_𝑗𝑖𝑒^T_𝑖(𝑡)Γ^TΓ𝑒_𝑖(𝑡)+𝑐𝛿₁^𝑁−1𝑙_{𝑖=𝑙+1𝑗=1}𝜆_max𝑃_𝑗𝐺_𝑗𝑖𝑒^T_𝑖(𝑡)Γ^TΓ𝑒_𝑖(𝑡). (26) Substituting (25)–(26) into (24) yields 𝐷⁺𝑉₁(𝑡)≤^𝑙_𝑖=1𝑒^T_𝑖(𝑡)−𝐴^T𝑃_𝑖−𝑃_𝑖𝐴𝑒_𝑖(𝑡)+𝜆_max𝑃_𝑖𝑒^T_𝑖(𝑡)×𝜖₁𝐼+𝜖₁⁻¹𝐿^T𝐿+𝑐𝛿₁^𝑁_𝑗=1𝐺_𝑖𝑗𝑒_𝑖(𝑡)+𝑐𝛿₁^𝑙−1𝑙_𝑖=1𝑗=1𝜆_max𝑃_𝑗𝐺_𝑗𝑖𝑒^T_𝑖(𝑡)Γ^TΓ𝑒_𝑖(𝑡)+𝑐𝛿₁^𝑁−1𝑙_{𝑖=𝑙+1𝑗=1}𝜆_max𝑃_𝑗𝐺_𝑗𝑖𝑒^T_𝑖(𝑡)Γ^TΓ𝑒_𝑖(𝑡)=^𝑙_𝑖=1𝑒^T_𝑖(𝑡)−𝐴^T𝑃_𝑖−𝑃_𝑖𝐴+𝜆_max𝑃_𝑖×𝜖₁𝐼+𝜖₁⁻¹𝐿^T𝐿+𝑐𝛿₁^𝑁_𝑗=1𝐺_𝑖𝑗+𝑐𝛿₁^𝑙−1_𝑗=1𝜆_max𝑃_𝑗𝐺_𝑗𝑖Γ^TΓ𝑒_𝑖(𝑡)+𝑐𝛿₁^𝑁−1𝑙_{𝑖=𝑙+1𝑗=1}𝜆_max𝑃_𝑗𝐺_𝑗𝑖𝑒^T_𝑖(𝑡)Γ^TΓ𝑒_𝑖(𝑡). (27) Define 𝛽_𝑖=𝑐𝛿₁⁻¹∑^𝑙_𝑗=1𝜆_max(𝑃_𝑗)|𝐺_𝑗𝑖|. Then it follows from condition (18) and Lemma 3 that 𝐷⁺𝑉₁(𝑡)≤^𝑙_𝑖=1𝑒^T_𝑖(𝑡)𝜆_𝑖𝑃_𝑖𝑒_𝑖(𝑡)+^𝑁_𝑖=𝑙+1𝛽_𝑖𝑒^T_𝑖(𝑡)𝑒_𝑖(𝑡)≤𝜆𝑉₁(𝑡)+𝛽𝑉₂(𝑡), (28) where 𝜆=max(𝜆_𝑖) and 𝛽=max(𝛽_𝑖𝜆_max(𝑄_𝑖⁻¹)).
Furthermore, for any 𝑡≥0, one can also get 𝐷⁺𝑉₂(𝑡)=^𝑁_𝑖=𝑙+1𝑒^T_𝑖̇(𝑡)𝑄_𝑖𝑒_𝑖(𝑡)+𝑒^T_𝑖(𝑡)𝑄_𝑖𝑒_𝑖̇(𝑡)=^𝑁_𝑖=𝑙+1𝑒^T_𝑖(𝑡)−𝐴^T𝑄_𝑖−𝑄_𝑖𝐴𝑒_𝑖(𝑡)+2𝑒^T_𝑖(𝑡)𝑄_𝑖𝑓𝑥_𝑖(𝑡)−𝑓(𝑥(𝑡))+2𝑐^𝑁_𝑗=1𝑒^T_𝑖(𝑡)𝑄_𝑖𝐺_𝑖𝑗Γ𝑒_𝑗(𝑡)≤^𝑁_𝑖=𝑙+1𝑒^T_𝑖(𝑡)−𝐴^T𝑄_𝑖−𝑄_𝑖𝐴𝑒_𝑖(𝑡)+𝜆_max𝑄_𝑖𝑒^T_𝑖(𝑡)×𝜖₂𝐼+𝜖₂⁻¹𝐿^T𝐿+𝑐𝛿₂^𝑁_𝑗=1𝐺_𝑖𝑗𝑒_𝑖(𝑡)+𝑐𝛿₂^𝑁−1_𝑗=𝑙+1𝜆_max𝑄_𝑗𝐺_𝑗𝑖𝑒^T_𝑖(𝑡)Γ^TΓ𝑒_𝑖(𝑡)+𝑐𝛿₂^𝑙−1𝑁_{𝑖=1𝑗=𝑙+1}𝜆_max𝑄_𝑗𝐺_𝑗𝑖𝑒^T_𝑖(𝑡)Γ^TΓ𝑒_𝑖(𝑡)=^𝑁_𝑖=𝑙+1𝑒^T_𝑖(𝑡)×−𝐴^T𝑄_𝑖−𝑄_𝑖𝐴+𝜆_max𝑄_𝑖×𝜖₂𝐼+𝜖₂⁻¹𝐿^T𝐿+𝑐𝛿₂^𝑁_𝑗=1𝐺_𝑖𝑗+𝑐𝛿₂^𝑁−1_𝑗=𝑙+1𝜆_max𝑄_𝑗𝐺_𝑗𝑖Γ^TΓ𝑒_𝑖(𝑡)+𝑐𝛿₂^𝑙−1𝑁_{𝑖=1𝑗=𝑙+1}𝜆_max𝑄_𝑗𝐺_𝑗𝑖𝑒^T_𝑖(𝑡)Γ^TΓ𝑒_𝑖(𝑡), (29) where 𝜖₂>0, 𝛿₂>0.
Defining 𝛾_𝑖=𝑐𝛿₂⁻¹∑^𝑁_𝑗=𝑙+1𝜆_max(𝑄_𝑗)|𝐺_𝑗𝑖|, it follows from (20) and (29) that 𝐷⁺𝑉₂(𝑡)≤−^𝑁_𝑖=𝑙+1𝜂_𝑖𝑒^T_𝑖(𝑡)𝑄_𝑖𝑒_𝑖(𝑡)+^𝑙_𝑖=1𝛾_𝑖𝑒^T_𝑖(𝑡)𝑒_𝑖(𝑡)≤−𝜂𝑉₂(𝑡)+𝛾𝑉₁(𝑡), (30) where 𝜂=min(𝜂_𝑖) and 𝛾=max(𝛾_𝑖𝜆_max(𝑃_𝑖⁻¹)), which implies 𝑉₂(𝑡)≤𝑒^−𝜂𝑡𝑉₂(0)+𝛾^𝑡₀𝑒^{−𝜂(𝑡−𝛼)}𝑉₁(𝛼)𝑑𝛼. (31) Then substituting (31) into (28) yields 𝐷⁺𝑉₁(𝑡)≤𝜆𝑉₁(𝑡)+𝛽𝑒^−𝜂𝑡𝑉₂(0)+𝛽𝛾^𝑡₀𝑒^{−𝜂(𝑡−𝛼)}𝑉₁(𝛼)𝑑𝛼,𝑡∈𝑡_𝑘,𝑡_𝑘+1. (32) On the other hand, when 𝑡=𝑡⁺_𝑘, it follows from (19) that 𝑉₁𝑡⁺_𝑘=^𝑙_𝑖=1𝑒^T_𝑖𝑡_𝑘𝐼+𝐸_𝑖𝑘^T𝑃_𝑖𝐼+𝐸_𝑖𝑘𝑒_𝑖𝑡_𝑘≤^𝑙_𝑖=1𝑒^T_𝑖𝑡_𝑘𝜇_𝑖𝑃_𝑖𝑒_𝑖𝑡_𝑘≤𝜇𝑉₁𝑡_𝑘, (33) where 𝜇=max(𝜇_𝑖), 𝑖=1,…,𝑙.
For any scalar 𝜖>0, define the following impulsive integrodifferential equation: 𝑉̇_𝜖(𝑡)=𝜆𝑉_𝜖(𝑡)+𝛽𝑒^−𝜂𝑡𝑉₂(0)+𝛽𝛾^𝑡₀𝑒^{−𝜂(𝑡−𝛼)}𝑉_𝜖(𝛼)𝑡𝑑𝛼+𝜖,𝑡∈_𝑘−1,𝑡_𝑘,𝑉_𝜖𝑡⁺_𝑘=𝜇𝑉_𝜖𝑡_𝑘,𝑡=𝑡_𝑘,𝑉_𝜖(0)=𝑉₁(0). (34) It then follows from Lemma 4 that 𝑉₁(𝑡)≤𝑉_𝜖(𝑡), for all 𝑡∈(0,𝑡_𝑘].
The solution to (34) can be expressed as follows: 𝑉_𝜖(𝑡)=𝑊(𝑡,0)𝑉_𝜖(0)+^𝑡₀𝑊()×𝑡,𝑠𝜖+𝛽𝑒^−𝜂𝑠𝑉₂(0)+𝛽𝛾^𝑠₀𝑒^{−𝜂(𝑠−𝛼)}𝑉_𝜖(𝛼)𝑑𝛼𝑑𝑠, (35) where 𝑊(𝑡,𝑠)(𝑡,𝑠≥0) is the Cauchy matrix of the linear impulsive system 𝑢̇(𝑡)=𝜆𝑢(𝑡),𝑡∈𝑡_𝑘−1,𝑡_𝑘,𝑢𝑡⁺_𝑘=𝜇𝑢𝑡_𝑘,𝑘=1,2,…,𝑡=𝑡_𝑘. (36)
Furthermore, noting that 𝜇<1 and 𝑡_𝑘+1−𝑡_𝑘≤𝛿, it follows from condition (21) that 𝑊(𝑡,𝑠) could be estimated as 𝑊(𝑡,𝑠)=𝑒^{𝜆(𝑡−𝑠)}_{𝑠<𝑡𝑘≤𝑡}𝜇≤𝑒^{𝜆(𝑡−𝑠)}𝜇^{((𝑡−𝑠)/𝛿)−1}=1𝜇𝑒^{−((1/𝛿)ln(1/𝜇)−𝜆)(𝑡−𝑠)}. (37) Defining 𝜆=(1/𝛿)ln(1/𝜇)−𝜆, one has 𝑉_𝜖(𝑡)≤1𝜇𝑒⁻𝜆_𝑡𝑉_𝜖(0)+1𝜇^𝑡₀𝑒⁻𝜆_{(𝑡−𝑠)}𝜖+𝛽𝑒^−𝜂𝑠𝑉₂(0+)𝑑𝑠𝛽𝛾𝜇^𝑡₀𝑒⁻𝜆_{(𝑡−𝑠)}^𝑠₀𝑒^{−𝜂(𝑠−𝛼)}𝑉_𝜖(𝛼)=1𝑑𝛼𝑑𝑠𝜇𝑒⁻𝜆_𝑡𝑉_𝜖(0)+𝜖𝜇𝜆1−𝑒⁻𝜆_𝑡+𝛽𝑉₂(0)𝜇𝑒𝜆−𝜂^−𝜂𝑡−𝑒⁻𝜆_𝑡+𝛽𝛾𝜇^𝑡₀^𝑡_𝛼𝑒⁻𝜆_{(𝑡−𝑠)}𝑒^{−𝜂(𝑠−𝛼)}𝑉_𝜖(𝛼)=1𝑑𝑠𝑑𝛼𝜇𝑒⁻𝜆_𝑡𝑉_𝜖(0)+𝜖𝜇𝜆1−𝑒⁻𝜆_𝑡+𝛽𝑉₂(0)𝜇𝑒𝜆−𝜂^−𝜂𝑡−𝑒⁻𝜆_𝑡+𝛽𝛾𝜇×𝜆−𝜂_𝑡0𝑒^{−𝜂(𝑡−𝛼)}−𝑒⁻𝜆_{(𝑡−𝛼)}𝑉_𝜖(𝛼)𝑑𝛼. (38) Noting from condition (21) that 𝜆>𝜂, then one gets 𝑉_𝜖(𝑡)≤1𝜇𝑒⁻𝜆_𝑡𝑉_𝜖(0)+𝜖𝜇𝜆+𝛽𝑉₂(0)𝜇𝑒𝜆−𝜂^−𝜂𝑡+𝛽𝛾𝜇𝜆−𝜂^𝑡0𝑒^{−𝜂(𝑡−𝛼)}𝑉_𝜖(𝛼)𝑑𝛼. (39) Defining 𝑢(𝑡)=𝑒^𝜂𝑡𝑉_𝜖(𝑡),𝛼(𝑡)=(𝑉_𝜖(0)/𝜇)+(𝛽𝑉₂(0)/𝜇(𝜆−𝜂))+(𝜖/𝜇𝜆)𝑒^𝜂𝑡, and 𝛽(𝑡)=(𝛽𝛾/𝜇(𝜆−𝜂))=𝜌, one obtains 𝑢(𝑡)≤𝛼(𝑡)+^𝑡₀𝛽(𝑠)𝑢(𝑠)𝑑𝑠,𝑡≥0. (40) Then by using Lemma 5, it is easy to get 𝑢(𝑡)≤𝑉_𝜖(0)𝜇+𝛽𝑉₂(0)𝜇𝑒𝜆−𝜂^𝜌𝑡+𝜖𝜂𝜇𝜆𝑒𝜂−𝜌^𝜂𝑡,𝑡≥0, (41) which implies 𝑉_𝜖(𝑡)≤𝑉_𝜖(0)𝜇+𝛽𝑉₂(0)𝜇𝑒𝜆−𝜂^{−(𝜂−𝜌)𝑡}+𝜖𝜂𝜇𝜆,𝜂−𝜌𝑡≥0. (42) Let 𝜖→0⁺, one can get 𝑉₁(𝑡)≤𝑒^{−(𝜂−𝜌)𝑡}𝑉₁(0)𝜇+𝛽𝑉₂(0)𝜇.𝜆−𝜂 (43) Then it follows from (31) that 𝑉₂(𝑡)≤𝑒^−𝜂𝑡𝑉₂(0)+𝑒^{−(𝜂−𝜌)𝑡}𝑉₁(0)𝜇+𝛽𝑉₂(0)𝜇𝛾𝜆−𝜂𝜌. (44) Furthermore, it follows from condition (21) that 𝜂>𝜌. Therefore together with (43) and (44), one can conclude that condition (17) is satisfied; that is, the entrained collective rhythms of multicellular systems (6) are achieved. The proof is thus completed.

Remark 8. The obtained result not only provides a new prospective to understand the interactions between the external stimuli and intrinsic physiological rhythms but also is potentially useful for the development of some medical devices. The result presented here is more effective in comparison with those in [38–41], where it is assumed that all the states are available for feedback purpose.
As a special case, if the positive matrices 𝑃_𝑖 and 𝑄_𝑖 in the Lyapunov function (22) are chosen as the identity matrix, the following simplified result could be readily obtained.

Corollary 9. For a given scalar 𝜇∈(0,1), if there exist scalars 𝜆_𝑖>0,  𝜇_𝑖>0,  𝑖=1,2,…,𝑙,𝜂_𝑖>0,  𝑖=𝑙+1,…,𝑁, and positive scalars 𝜖₁,  𝜖₂,  𝛿₁,  and 𝛿₂ such that −𝐴^T−𝐴+𝜖₁𝐼+𝜖₁⁻¹𝐿^T𝐿+𝑐𝛿₁^𝑁_𝑗=1𝐺_𝑖𝑗+𝑐𝛿₁^𝑙−1_𝑗=1𝐺_𝑗𝑖Γ^TΓ≤𝜆_𝑖,𝐼+𝐸_𝑖𝑘^𝑇𝐼+𝐸_𝑖𝑘≤𝜇_𝑖, (45) for 𝑖=1,…,𝑙 and −𝐴^T−𝐴+𝜖₂𝐼+𝜖₂⁻¹𝐿^T𝐿+𝑐𝛿₂^𝑁_𝑗=1𝐺_𝑖𝑗+𝑐𝛿₂^𝑁−1_𝑗=𝑙+1𝐺_𝑗𝑖Γ^TΓ≤−𝜂_𝑖, (46) for 𝑖=𝑙+1,…,𝑁, and, for any impulsive time sequence {𝑡_𝑘} satisfying 𝛿∶=sup_𝑘𝑡_𝑘+1−𝑡_𝑘<ln1/𝜇/,𝜂+𝜆+𝛽𝛾𝜂𝜇 (47) for 𝑘=0,1,2,…, where 𝜇=max(𝜇_𝑖),  𝜆=max(𝜆_𝑖),  𝑖=1,…,𝑙, and 𝜂=min(𝜂_𝑖),  𝑖=𝑙+1,…,𝑁, then the entrained collective rhythms of multicellular systems (6) are achieved.

4. Numerical Example

In this section, a synthetic multicellular system composed of 𝑁 Goodwin oscillators [33] is employed to illustrate the effectiveness of the proposed control strategy.

4.1. Goodwin Oscillators

The Goodwin oscillator is a genetic network with negative feedback formed in a cyclic way [33], where metabolites repress the enzymes which are essential for their own synthesis by inhibiting the transcription of the molecule DNA to messenger RNA (mRNA) (see [42] for more details). The regulatory scheme of the Goodwin oscillator can be shown in Figure 1.

Figure 1 

Regulatory scheme of the Goodwin oscillator.

A modified model reflecting the above regulation process is given by 𝑑𝑚=𝑢𝑑𝑡𝑘₁+𝑝^𝛼−𝑎𝑚,𝑑𝑒𝑑𝑡=𝑣𝑚−𝑏𝑒,𝑑𝑝𝑑𝑡=𝑤𝑒−𝑐𝑝𝑘₂,+𝑝 (48) where 𝑚, 𝑒, and 𝑝 are the concentrations of mRNA, the enzyme, and the product of the reaction of the enzyme and a substrate, respectively. 𝑎, 𝑏, and 𝑐 are the degradation rates of each component, respectively. 𝑢, 𝑣, and 𝑤 denote the rates of transcription, translation, and catalysis, respectively. 𝑘₁ and 𝑘₂ are two positive constants. 𝛼 is the Hill coefficient denoting the cooperativity of the end product repression. It should be pointed out that model (46) is slightly different from the model in [33]. By changing the linear degradation form of the product to Michaelis-Menten form, the limit cycle oscillations can occur for a lower value of the Hill coefficient 𝛼 (see [42] for details).

The parameters are chosen as 𝑎=𝑏=𝑐=𝑣=𝑤=0.1, 𝑢=1, and 𝑘₁=𝑘₂=0.1, and the Hill coefficient 𝛼 is taken as 𝛼=4. Given 4 random initial conditions, the simulation results are recorded in Figures 2 and 3. Figure 2 shows the time response of every products, and Figure 3 shows the limit cycle of the oscillators in phase space.

Figure 2 

Time response of four oscillators.

Figure 3 

Limit cycle of the oscillators in phase space.

4.2. Multicellular System Model

Without loss of generality, we perform the simulation study on the scale-free network structure, which is assumed to obey the scale-free distribution of the Barabási-Albert (BA) model [43]. The parameters are given as 𝑚₀=𝑚=5 and 𝑁=60. Figure 4 is the generated BA network graph.

Figure 4 

BA scale-free network graph.

The inner coupling matrix Γ is given by Γ=diag(1,1.2,1), and the regulation function in Michaelis-Menten form satisfies condition (2). Assume that only the first 20 nodes are able to accept the external inputs. The parameters are chosen as 𝑐=4, 𝜖₁=0.03, and 𝜖₂=0.01. The impulse magnitude is given as 𝐸_𝑖𝑘=−1.8, and impulse interval is chosen as 𝛿=0.4. It follows from Corollary 9 that the entrained collective behaviors can be achieved. Figures 5, 6, and 7 show the rhythmic errors of the multicellular systems.

Figure 5 

Rhythmic errors 𝑒𝑖1 in BA network.

Figure 6 

Rhythmic errors 𝑒𝑖2 in BA network.

Figure 7 

Rhythmic errors 𝑒𝑖3 in BA network.

It is noted that since only 20 measurable cells in the network are subject to the external stimuli, the approaches in [38–41], which need the information of all the states, cannot be applied in this case.

5. Conclusion

In this paper, the entrained collective rhythms of multicellular systems have been investigated. It is shown that the entrained collective behavior can be achieved via impulsive control even when only partial states of multicellular systems are available. With the help of the newly developed impulsive integrodifferential inequality, the sufficient conditions are derived to ensure the entrained collective rhythms of multicellular systems. A synthetic multicellular system is finally used to illustrate the effectiveness of the developed impulsive control strategy.

Acknowledgments

The work was supported in part by the National Natural Science Foundation of China under Grant 61104045, in part by Technology Foundation for Selected Overseas Chinese Scholar, Ministry of Personnel of China, and in part by the Fundamental Research Funds for the Central Universities of China under Grant 2012B03514.