How to desynchronize quorum-sensing networks

In this paper we investigate how so-called quorum-sensing networks can be desynchronized. Such networks, which arise in many important application fields, such as systems biology, are characterized by the fact that direct communication between network nodes is superimposed to communication with a shared, environmental variable. In particular, we provide a new sufficient condition ensuring that the trajectories of these quorum-sensing networks diverge from their synchronous evolution. Then, we apply our result to study two applications.

Figure 1

Geometric interpretation of Definition t1. Two trajectories, $x\left(t\right)$ [with initial condition $x\left(t_0\right)$] and $x^{*}\left(t\right)$ [with initial condition $x^{*}\left(t_0\right)$] are shown. The open sets ${\mathcal{B}}_{\delta }(x\left(t\right))$ define, over time, an open bundle (in blue in the color figure online) around $x\left(t\right)$. The two trajectories have nearby initial conditions, i.e., $x^{*}\left(t_0\right)$ belongs to ${\mathcal{B}}_{\delta }(x\left(t_0\right))$. The distance between trajectory $x\left(t\right)$ and $x^{*}\left(t\right)$ increases and this causes $x^{*}\left(t\right)$ to exit from the bundle defined by ${\mathcal{B}}_{\delta }(x\left(t\right))$. Note that Definition t1 does not provide any insight on how the distance $\left|x^{*}\left(t\right)-x\left(t\right)\right|$ evolves once $x^{*}\left(t\right)$ is outside of the bundle.

Figure 2

Networks considered in this paper. Network nodes interact with each other and with a shared environmental variable. Both network nodes and the shared variable are modeled via a set of ODEs.

Figure 3

A motivating application, where a sensor network estimates a quantity of interest and sends the estimate to a remote service.

Figure 4

Time evolution for the small-world network considered in Sec. 5a. Time is on the $x$ axis and $x_i$'s are on the $y$ axis. In the top panel the network nodes' behavior is shown for $\overline{q}=2.5$: such a panel shows that all the nodes properly sense the quantity of interest. In the right panel, the nodes' time behavior is instead shown for $\overline{q}=0.25$. In such a panel, nodes are not able to sense the quantity of interest as no agreement is reached. Initial conditions for the network nodes are taken from a standard distribution.

Figure 5

Order parameter as a function of ${\gamma }_1$. Values of ${\gamma }_1$ on the $x$ axis and the order parameter $R$ on the $y$ axis. The increase of this parameter causes a loss of synchronization for both Network 1 (top panel) and Network 2 (bottom panel). Note that Network 1 starts to desynchronize after Network 2, thus confirming our theoretical predictions.

Figure 6

Time is on the $x$ axis and $x_i$'s on the $y$ axis. The time evolution for Network 1 (top panel) and Network 2 (bottom panel) is shown when ${\gamma }_1=10$. The two panels show that two groups (or clusters) of synchronized nodes emerge.