An Exploration of Dynamics and Bifurcation in Networks

By Martin Golubitsky and Ian Stewart

The following is a brief description from the authors of Dynamics and Bifurcation in Networks: Theory and Applications of Coupled Differential Equations, which was published by SIAM in 2023. The book acknowledges recent scientific interest in network-based modeling, addresses the past two decades of developments in the formalism of network dynamics, and investigates the influence of a network’s architecture and symmetry on the behavior of dynamical systems.

Dynamics and Bifurcation in Networks: Theory and Applications of Coupled Differential Equations examines the qualitative features of solutions to admissible systems of ordinary differential equations (ODEs) that are supported by a fixed network, i.e., a directed graph or digraph (more generally, we can assign “types” to nodes and edges that the ODE must preserve). Examples of simple networks include a bidirectional ring of two identical nodes and a unidirectional ring of three nonidentical nodes (see Figure 1).

The admissible systems for these two networks are as follows:

\[\begin{equation}
(\text{a}) \begin{array}{rcl}
\dot{x} & = & f(x,y) \\
\dot{y} & = & f(y,x)
\end{array}
\qquad
(\text{b}) \begin{array}{rcl}
\dot{x} & = & f(x,y) \\
\dot{y} & = & g(y,z)\\
\dot{z} & = & h(z,x)
\end{array}
\end{equation}.\tag1\]

The two-node system in \((1\textrm{a})\) is homogeneous because the \(\dot{x}\) and \(\dot{y}\) node functions are identical, as indicated by the matching node and edge symbols. The three-node system in \((1\textrm{b})\) is fully inhomogeneous because the \(\dot{x}\), \(\dot{y}\), and \(\dot{z}\) node functions are generally different, as indicated by the distinct node and edge symbols. The three-node digraph implies that the time evolution of the \(x\) node depends on the states of the \(x\) and \(y\) nodes but not of the \(z\) node, and so on.

We claim that differences frequently exist between the generic features of the solutions to admissible systems that correspond with individual networks. Here, “generic” refers to typical behavior in the world of admissible ODEs for the digraph in question. For additional context, we know that there are more than 1.5 million six-node digraphs. Even if our claim about different dynamics for different networks is only partially valid, the corresponding mathematical problem is huge.

We make two assertions: (i) Applications often lead to admissible systems of a fixed network, and (ii) admissibility often leads to generically interesting mathematics. Here, we use Hopf bifurcation from dynamical systems theory to describe an example of each assertion. Suppose that a stable equilibrium loses its stability when a pair of simple, purely imaginary eigenvalues of the Jacobian matrix cross zero with nonzero speed. Hopf bifurcation affirms that a unique branch of small-amplitude periodic solutions emanates from the bifurcation point.

The two-node example implies that periodic solutions with period \(T\)—where \(y(t) = x(t+T/2)\)—are unsurprising. Also, two nodes \(i\) and \(j\) in a given network \(\mathcal{G}\) are equivalent if a path connects \(i\) to \(j\) and another path connects \(j\) to \(i\); a component of \(\mathcal{G}\) is called an equivalence class. The subsequent theorem illustrates the interaction between networks and dynamics. Let \(\mathcal{H}\) be an equivalence class of \(\mathcal{G}\). A periodic solution is of type \(\mathcal{H}\) if \(x(t)\) oscillates in every node within \(\mathcal{H}\) or downstream from \(\mathcal{H}\), but is stationary in all remaining nodes. The theorem is thus as follows: For each component \(\mathcal{H}\) in the fully inhomogeneous network \(\mathcal{G}\), a certain type of Hopf bifurcation leads generically to \(\mathcal{H}\)-type periodic solutions.

Next, we sketch an application to binocular rivalry — a psychophysics experiment wherein a subject is shown two different images. More precisely, the subject’s right eye is exposed to one image and their left eye is exposed to the other; the individual then records the image that they perceive (known as the percept) over time. The subject typically perceives one image and then the other, with continuous alternation throughout the experiment. Although the reported alternation is not usually periodic, binocular rivalry models often focus on finding a periodic alternation of the images. The model equations typically take the form

\[\begin{eqnarray} \dot{a}^E & = & f(a^E, a^H, b^E) \\ \dot{a}^H & = & g(a^E, a^H) \\ \dot{b}^E & = & f(b^E, b^H, a^E) \\  \dot{b}^H & = & g(b^E, b^H), \end{eqnarray}\tag2\]

where each two-dimensional node \(a\) and \(b\) has an activity variable \(*^E\) and a fatigue variable \(*^H\).

In this model, the subject perceives the \(a\) percept when \(a^E>b^E\) and the \(b\) percept when \(a^E<b^E\). States where \(a^E=b^E\) are called fusion states; the model does not choose a percept in this case. Many studies that examine these equations’ rich dynamics are available in the literature [1, 2, 5, 7].

One particular study involves two specific binocular rivalry experiments [4]. In the first experiment, one eye is shown an image of a monkey and the other is shown an image of a jungle scene with overlaid text (see Figure 2). Based on the network in \((1 \textrm{a})\), a perceived alternation does indeed occur between the two images.

In the second experiment, three lines subdivide the rectangular monkey-text images into six wedges that are then reassembled into two scrambled rectangles (see Figure 3). This modified exercise yields a curious result. While subjects do perceive the expected alternation between the scrambled images, they also surprisingly perceive an alternation between the original, unscrambled images.

Why does this happen? Hugh Wilson proposed a homogeneous network model in which this outcome—i.e., alternation between both the standard and scrambled monkey-text images—is unsurprising [6, 7]. Chapter 23 of Dynamics and Bifurcation in Networks, titled “Binocular Rivalry and Visual Illusions,” explores this question in more depth.

Dynamics and Bifurcation in Networks provides an extensive introduction to the examples, theory, and applications of network dynamics. Each chapter commences with an introductory section that helps readers isolate parts of the material that are not necessarily essential on a first perusal. At the beginning of the book, a 16-page preface orients readers to the more extensive material.

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References
[1] Curtu, R. (2010). Singular Hopf bifurcations and mixed-mode oscillations in a two-cell inhibitory neural network. Physica D: Nonlin. Phenom., 239(9), 504-514.
[2] Curtu, R., Shpiro, A., Rubin, N., & Rinzel, J. (2008). Mechanisms for frequency control in neuronal competition models. SIAM J. Appl. Dyn. Syst., 7(2), 609-649.
[3] Diekman, C.O., Golubitsky, M., & Wang, Y. (2013). Derived patterns in binocular rivalry networks. J. Math. Neurosci., 3(1), 6.
[4] Kovács, I., Papathomas, T.V., Yang, M., & Fehér, Á. (1996). When the brain changes its mind: Interocular grouping during binocular rivalry. Proc. Natl. Acad. Sci., 93(26), 15508-15511.
[5] Laing, C.R., & Chow, C.C. (2002). A spiking neuron model for binocular rivalry. J. Comput. Neurosci., 12(1), 39-53.
[6] Wilson, H.R. (2007). Minimal physiological conditions for binocular rivalry and rivalry memory. Vision Res., 47(21), 2741-2750.
[7] Wilson, H.R. (2009). Requirements for conscious visual processing. In M. Jenkin and L. Harris (Eds.), Cortical mechanisms of vision (pp. 399-417). Cambridge, U.K.: Cambridge University Press.

	Martin Golubitsky is an emeritus professor of mathematics at the Ohio State University. He is the founding editor-in-chief of the SIAM Journal on Applied Dynamical Systems and a past president of SIAM.
	Ian Stewart is an emeritus professor of mathematics at the University of Warwick in the U.K. He has published more than 200 research papers, authored many popular mathematics books, and is a Fellow of the Royal Society.