Symbolic Dynamics of Biological Feedback Networks

We formulate general rules for a coarse graining of the dynamics, which we term “symbolic dynamics,” of feedback networks with monotonic interactions, such as most biological modules. Networks which are more complex than simple cyclic structures can exhibit multiple different symbolic dynamics. Nevertheless, we show several examples where the symbolic dynamics is dominated by a single pattern that is very robust to changes in parameters and is consistent with the dynamics being dictated by a single feedback loop. Our analysis provides a method for extracting these dominant loops from short time series, even if they only show transient trajectories.

Figure 1

Simple example of the use of symbolic dynamics. (a) (left) Scheme of the network. We represent activation by a normal arrow, and repression by a barred arrow. (right) Corresponding transition network. We removed for clarity the symbols ($+-+$) and ($-+-$) which have no incoming links. (b) Dynamics of the 3 variables as a function of time with a weak cross-link ($a=10$, see text), showing the transition cycle in solid arrows in (a). Inset shows the same on a log scale. (c) Dynamics for a stronger cross-link ($a=50$, see text) which includes the transitions shown by dashed arrows in (a), zoomed in the inset. In all plots $x_1$ is blue, $x_2$ is green, and $x_3$ is red.

Figure 2

Network of two coupled two-species oscillators. (a) Structure of the network. (b) The transition network for this 3-node system. Black arrows indicate all the allowed transitions. Blue arrows are the transitions actually observed in the HP system, and red arrows are the transitions observed in the BHS model (see text). Dashed arrows indicate “kicks,” i.e., transitions which are not observed close to the Hopf bifurcation.

Figure 3

Two dimensional projection of the attractor of the system of Eq. (3) for different values of the control parameter $r=2.0$ (top left), $r=2.6$ (top right), $r=3.0$ (bottom left), $r=3.3$ (bottom right).

Figure 4

Bifurcation diagrams. (top) HP model (3), minima of $x_2$ plotted versus $r$. (bottom) BHS model (4), maxima of $x_1$ versus $c$. In both plots, red dots indicate the appearance of kicks in the trajectory and symbolic dynamics (see text).