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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.

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 [

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 [

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 [

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

To make it easy for the readers, let us start from a single cell model of the form

It is known that many biological models can be represented by (

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

The regulatory function

Consider multicellular systems with cell-to-cell communication mechanism described as follows:

Suppose only

Then the impulsive-controlled multicellular systems with partial states can be described by the following impulsive differential equation:

Defining

Then the problem of entrained collective rhythms is to design the partial impulsive controller (

If

For positive scalars

Firstly, we prove

Suppose

Let

The multicellular system (

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

For a given scalar

Consider the following Lyapunov function:

For any

Furthermore, for any

Defining

For any scalar

The solution to (

Furthermore, noting that

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 [

As a special case, if the positive matrices

For a given scalar

In this section, a synthetic multicellular system composed of

The Goodwin oscillator is a genetic network with negative feedback formed in a cyclic way [

Regulatory scheme of the Goodwin oscillator.

A modified model reflecting the above regulation process is given by

The parameters are chosen as

Time response of four oscillators.

Limit cycle of the oscillators in phase space.

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 [

BA scale-free network graph.

The inner coupling matrix

Rhythmic errors

Rhythmic errors

Rhythmic errors

It is noted that since only 20 measurable cells in the network are subject to the external stimuli, the approaches in [

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.

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.