Chaos in a continuous-time Boolean network

TL;DR

This paper investigates chaos in continuous-time Boolean networks, demonstrating that even simple 4-dimensional systems with Boolean interactions can exhibit complex, chaotic dynamics similar to Smale horseshoes.

Contribution

It provides a concrete example of chaos in a continuous-time Boolean network, linking discrete map properties to continuous dynamics and revealing complex behaviors in low-dimensional systems.

Findings

Presence of chaotic dynamics in a 4D Boolean network

Identification of a Smale horseshoe-like structure

Chaotic behavior in systems with four or more variables

Abstract

Continuous-time systems with switch-like behaviour occur in chemical kinetics, gene regulatory networks and neural networks. Networks with hard switching, as a limiting case of smooth sigmoidal switching, retain the richest possible range of behaviors but are mathematically more tractable. The form of an underlying discrete (fractional-linear) map encodes information on existence, stability and exact periods of periodic orbits. In richly connected structures with four or more variables, aperiodic behaviour can occur. We investigate a simple 4-dimensional example with Boolean interaction terms in which a Smale horseshoe-like object reveals chaotic dynamics.

Tables

orthant ($\widetilde{𝐲}$)	focal pt. ($𝐅$)
0 0 0 0	$-1$	$1$	$1$	$1$
0 0 0 1	$-1$	$1$	$-1$	$1$
0 0 1 0	$1$	$-1$	$1$	$-1$
0 0 1 1	$1$	$-1$	$-1$	$-1$
0 1 0 0	$-1$	$1$	$1$	$1$
0 1 0 1	$-1$	$1$	$1$	$1$
0 1 1 0	$1$	$-1$	$1$	$-1$
0 1 1 1	$1$	$-1$	$1$	$-1$
1 0 0 0	$-1$	$1$	$-1$	$-1$
1 0 0 1	$-1$	$1$	$-1$	$-1$
1 0 1 0	$1$	$1$	$-1$	$-1$
1 0 1 1	$1$	$-1$	$-1$	$-1$
1 1 0 0	$-1$	$1$	$-1$	$1$
1 1 0 1	$-1$	$1$	$1$	$1$
1 1 1 0	$1$	$1$	$-1$	$1$
1 1 1 1	$1$	$-1$	$1$	$1$

Equations

$$\dot{y_i}=-y_i+F_i({\tilde{y}}_1,{\tilde{y}}_2,\dots ,{\tilde{y}}_n),i=1,\dots ,n,$$

$$\tilde{y}_{i}=\left\{\begin{array}[]{ll}0&\mbox{if $y_{i}<0$}\\ 1&\mbox{if $y_{i}>0$}\end{array}\right.\,.$$

$$\mathbf{y}(t)=\mathbf{f}+(\mathbf{y}(0)-\mathbf{f})e^{-t},$$

$${\mathbf{y}}^{(k+1)}=M^{(k)}{\mathbf{y}}^{(k)}=\frac{B^{(k)}{\mathbf{y}}^{(k)}}{1+⟨{\psi }^{(k)},{\mathbf{y}}^{(k)}⟩},B^{(k)}=I-\frac{{\mathbf{f}}^{(k)}{\mathbf{e}}_j^{\prime }}{f_j^{(k)}},{\psi }^{(k)}=\frac{-{\mathbf{e}}_j}{f_j^{(k)}},$$

$$M\mathbf{y}=\frac{A\mathbf{y}}{1+⟨\varphi ,\mathbf{y}⟩},$$

$$R_{i,\cdot }=-\frac{{\mathbf{e}}_i^{\prime }}{f_i^{(k)}}{B}^{(k)}{B}^{(k-1)}\dots B^{(0)}.$$

$${\mathbf{y}}^{\ast }=\frac{(\lambda -1)\mathbf{v}}{⟨\varphi ,\mathbf{v}⟩}$$

$$\begin{array}[]{lcl}\dot{y}_{1}&=&-y_{1}+2[\tilde{y}_{3}]-1\\ \dot{y}_{2}&=&-y_{2}+2[1-\tilde{y}_{3}+\tilde{y}_{1}\tilde{y}_{3}-\tilde{y}_{1}\tilde{y}_{3}\tilde{y}_{4}]-1\\ \dot{y}_{3}&=&-y_{3}+2[(1-\tilde{y}_{1})(1-\tilde{y}_{4})+\tilde{y}_{2}\tilde{y}_{4}]-1\\ \dot{y}_{4}&=&-y_{4}+2[(1-\tilde{y}_{1})(1-\tilde{y}_{3})+\tilde{y}_{1}\tilde{y}_{2}]-1\,.\end{array}$$

$$\begin{array}[]{l}0101\rightarrow 0111\rightarrow 1111\rightarrow 1011\rightarrow{\bf 1001}\rightarrow 1000\rightarrow 1100\rightarrow 1101\quad\mbox{ and}\\ 0101\rightarrow 0111\rightarrow 1111\rightarrow 1011\rightarrow{\bf 1010}\rightarrow 1000\rightarrow 1100\rightarrow 1101\,.\end{array}$$

$$A_{0}=\left(\begin{array}[]{rrr}1&0&0\\ -2&5&2\\ 0&2&1\end{array}\right)\,,\quad\phi_{0}=(4,-4,0)^{T}\,,$$

$$A_{1}=\left(\begin{array}[]{rrr}1&-2&-2\\ -2&-3&-6\\ 0&-2&-3\end{array}\right)\,,\quad\phi_{1}=(4,-4,0)^{T}\,.$$

$${\lambda }_1\approx 5.8284,{\lambda }_2=1.0000,{\lambda }_3\approx 0.1716$$

$${\bf v}_{1}\approx\left(\begin{array}[]{r}0.0000\\ 0.7071\\ 0.2929\end{array}\right)\,,\quad{\bf v}_{2}=\left(\begin{array}[]{r}0.5000\\ 0.0000\\ 0.5000\end{array}\right)\,,\quad{\bf v}_{3}\approx\left(\begin{array}[]{r}0.0000\\ -0.2929\\ 0.7071\end{array}\right)\,,$$

$${\lambda }_1\approx -6.8709,{\lambda }_2\approx 1.9457,{\lambda }_3\approx -0.0748$$

$${\bf v}_{1}\approx\left(\begin{array}[]{r}0.2026\\ 0.5257\\ 0.2716\end{array}\right)\,,\quad{\bf v}_{2}\approx\left(\begin{array}[]{r}0.4728\\ -0.3754\\ 0.1518\end{array}\right)\,,\quad{\bf v}_{3}\approx\left(\begin{array}[]{r}-0.2590\\ -0.4401\\ 0.3009\end{array}\right)\,.$$

Full text

Along a cycle on the $n$-cube, there may be branching nodes, i.e., nodes with more than one outgoing edge. These correspond to orthants from which trajectories can exit by more than one boundary hyperplane, depending on which variable reaches zero first. Alternate exit variables impose constraints on the region of an orthant boundary that maps forwards through a specified sequence of boundaries. These constraints take the form of linear inequalities, and the restricted regions are the interiors of ‘proper cones’ (Berman & Plemmons [1], p.6).

[TABLE]

We allow equality, $R{\bf y}=0$, (trajectories for which two variables cross simultaneously) as limiting cases. Many of the inequalities generated by Equation 6 will be redundant and can be weeded out in computation.

The domain of definition of $M$ is only $C\subset{\cal O}$. Trajectories starting outside of $C$, but in ${\cal O}$, eventually branch away from the given cycle. Note also that $M$ maps $C$ into ${\cal O}$, not necessarily into $C$. However, a fixed point of the map lying inside $C$ continues to return and corresponds to a periodic orbit for the differential equations. If $C$ is empty, no periodic orbit corresponding to this $n$-cube cycle exists.

[TABLE]

is a fixed point of $M$, unique in the span of ${\bf v}$. If $\lambda=1$, then the only fixed point in the span of ${\bf v}$ is ${\bf 0}$.

4. APERIODIC BEHAVIOR

Aperiodic behavior in Glass networks appears common for large $n$. Numerical evidence for ergodicity via an invariant measure for a 6-dimensional network was given by Lewis & Glass [10]. For this and a similar 6-dimensional network discussed by Edwards et al [3], aperiodic behavior appeared for certain values of a parameter. Mestl et al [12] showed that chaos cannot exist in 3-dimensional Glass networks, but a 4-dimensional network with a special set of parameters was shown by Mestl et al [13] to have aperiodic dynamics, though it was not proven that there was a chaotic attractor. None of these examples were Boolean Glass networks and it was not known whether 4-dimensional networks of this class could exhibit aperiodicity.

Numerical experiments that we performed on randomly generated 4-dimensional Boolean Glass networks produced examples for which no fixed point or periodic orbit was detected. We investigate one of these, defined by the interaction function, ${\bf F}$, in Fig. 1. One way to express this network as a system of ODEs is as follows:

[TABLE]

A projection of an example 4-dimensional trajectory is shown in Fig. 2a.

Consider the two cycles,

[TABLE]

Both are feasible and starting from the $(0+-+)$ boundary (i.e., the boundary between 1101 and 0101) they have return maps $M_{0}$ and $M_{1}$ defined respectively by

[TABLE]

[TABLE]

Their eigenvalues ($\lambda_{i}$) and corresponding eigenvectors (${\bf v}_{i}$) are for $A_{0}$,

[TABLE]

[TABLE]

and for $A_{1}$,

[TABLE]

[TABLE]

Neither $A_{0}$ nor $A_{1}$ has its dominant eigenvector in $(+-+)$, and therefore, neither cycle has a stable periodic orbit. Fig. 2b shows the returning cones for these two cycles in the $(0+-+)$ boundary. In order to depict these 3-dimensional objects in a plane figure, we have projected the cones onto the plane $y_{2}-y_{3}+y_{4}=1$, which is the part of the unit ${\it l}^{1}$ ball in ${\bf R}^{3}$ that lies in the $(+-+)$ octant, and then plotted $y_{3}$ vs. $y_{4}$. Trajectories starting in the region labelled $C_{0}$ follow the first cycle above, $M_{0}$, and return to the $(0+-+)$ boundary. Similarly, trajectories starting in $C_{1}$ follow $M_{1}$.

The images of these two regions under their respective maps are also shown in Fig. 2b. The marked point at $\left(0,\frac{1}{2}\right)\in C_{0}$ represents the eigenvector ${\bf v}_{2}$ of $A_{0}$ ($\lambda_{2}>0$) so the ray through ${\bf v}_{2}$ in ${\bf R}^{3}$ is invariant under $M_{0}$ (and thus so is its projection in the figure), but since the corresponding eigenvalue is 1, no non-zero point on the ray is actually fixed and the origin attracts. The stretching and contraction in the directions of the other two eigenvectors (which are perpendicular and lie in the $y_{3}$–$y_{4}$ plane) are clearly visible. Since both eigenvalues are positive, there is no inversion of the image in either direction. Note that the eigenvector ${\bf v}_{3}$ with projected coordinates $(\frac{\sqrt{2}}{2}-1,\frac{\sqrt{2}}{2})\approx(-0.2929,0.7071)$ lies outside $C_{0}$.

The returning region $C_{1}$ for map $M_{1}$ is also subject to stretching and contracting in similar directions. The eigenvector ${\bf v}_{2}$ associated with eigenvalue $\lambda_{2}\approx 1.9457$ lies inside $C_{1}$, so the ray through this point is invariant, and there is an unstable fixed point of $M_{1}$ on this ray at $(0.1318,-0.1046,0.0423)$, which appears on the projection at $(-0.3754,0.1518)$. The period of the corresponding unstable orbit is $\log(\lambda_{2})\approx 0.6656$. The other two eigenvalues are negative, so in the other two directions (in which the $C_{1}$ region is stretched and contracted) we also have inversion.

The combined map, defined by $M_{0}$ and $M_{1}$, restricted to the projected plane, therefore contains something similar to a Smale horseshoe [2]. It is not exactly topologically equivalent to the horseshoe, but retains many of its properties. Points getting mapped out of $C_{0}\cup C_{1}$ will go elsewhere but there is nevertheless a Cantor set, $\Lambda$, in the projected plane, consisting of points whose trajectories remain in $C_{0}\cup C_{1}$ both forwards and backwards in time, and an infinite set of unstable periodic points. The main difference from the Smale horseshoe is that no points in the region $M_{1}(C_{1})\cap C_{0}$ are mapped (by $M_{0}$) into $C_{0}$ (this is easy to check by finding the images of the vertices of $M_{1}(C_{1})\cap C_{0}$ and joining them up by straight lines to find the images of the boundary edges, since straight lines are mapped to straight lines, even when projected). Thus, no points of $\Lambda$, aside from $\left(\frac{1}{2},0,\frac{1}{2}\right)$ itself, lie in $M_{0}(C_{0})\cap C_{0}$ and trajectories of periodic points never follow the $M_{0}$ cycle twice in a row, i.e., their symbolic trajectories do not contain the string ‘00’. Compositions of $M_{0}$ and $M_{1}$ for which $M_{0}$ is not repeated twice produce unstable periodic orbits. For example, the fixed points corresponding to $M_{0}M_{1}$ and $M_{1}M_{0}$ are marked in Fig. 2b.

In order to deal with the radial direction that is suppressed in the above projection, we need only confirm that trajectories from points in the Cantor set, $\Lambda$, (or on the rays through points of $\Lambda$) do not converge to the origin. In this case convergence in the radial direction ensures that asymptotically we can ignore the radial component and we approach the 2-dimensional chaotic dynamics of $\Lambda$.

Boundedness away from the origin can be shown by finding a neighbourhood of the origin in which points on rays through $\Lambda$ always move away from the origin under the map. Simple but tedious calculations show that for each of the corners, $Q_{i},\,i=1,\ldots,12$, of the three regions $M_{0}(C_{0})\cap C_{1}$, $M_{1}(C_{1})\cap C_{1}$ and $M_{1}(C_{1})\cap C_{0}$, the ${\it l}^{1}$ norm of the appropriate map, $M_{j}$, satisfies $\|M_{j}(kQ_{i})\|_{1}>\|kQ_{i}\|_{1}=k$ if $0<k<\frac{3}{22}$. Since planes are mapped to planes by each of $M_{0}$ and $M_{1}$, the same is true for rays through any point in the interior of one of these three regions (which contain $\Lambda$). Thus, points closer to the origin than $\|{\bf y}\|_{1}=\frac{3}{22}$ move away and there is no possibility of convergence to the origin on any ray through $\Lambda$.

5. DISCUSSION

We have shown that chaotic dynamics exist for the network of Fig. 1. We have not shown that orbits in $\Lambda$ are attracting and, in fact, numerical evidence suggests that they are not. When these equations are integrated, trajectories do return repeatedly to the orthant boundary $(0+-+)$, but have itineraries including several other cycles besides $M_{0}$ and $M_{1}$. These other cycles have returning cones in the gaps left by $C_{0}$ and $C_{1}$. Thus, the dynamics are actually more complicated than suggested by the analysis above.

It is surprising that complex dynamics are possible in such a simple network of only 4 variables with Boolean interactions. It certainly is not possible in 4-element discrete-time switching networks. Nevertheless, the techniques discussed here allow considerable progress in analysis of these Glass networks. An important unsolved question is to find necessary or sufficient conditions on the connection structure for chaotic dynamics to occur.

ACKNOWLEDGEMENTS

This work was partially supported by grants from the University of Victoria and the Natural Sciences and Engineering Research Council of Canada.

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