Turing patterns on discrete topologies: from networks to higher-order structures

TL;DR

This work surveys the extension of Turing pattern theory from continuous reaction-diffusion systems to discrete supports, first via networks and later through higher-order topologies. It unifies discretization-based dispersion analysis, directed/non-normal effects, multiplex and temporal networks, and the Master Stability Function framework, and then extends the theory to higher-order interactions with hypergraphs, simplicial complexes, and topological signals using boundary, coboundary, and Dirac operators. Key contributions include explicit dispersion-formulations on networks, conditions for diffusion-driven instabilities in diverse topologies, and analytical as well as conceptual frameworks for pattern formation on higher-order structures. The study broadens the applicability of Turing-type self-organization to neuroscience, ecology, and complex systems, offering new analytical tools and guiding future research on directed topologies, higher-order diffusion, and topological pattern formation.

Abstract

Nature is a blossoming of regular structures, signature of self-organization of the underlying microscopic interacting agents. Turing theory of pattern formation is one of the most studied mechanisms to address such phenomena and has been applied to a widespread gallery of disciplines. Turing himself used a spatial discretization of the hosting support to eventually deal with a set of ODEs. Such an idea contained the seeds of the theory on discrete support, which has been fully acknowledged with the birth of network science in the early 2000s. This approach allows us to tackle several settings not displaying a trivial continuous embedding, such as multiplex, temporal networks, and, recently, higher-order structures. This line of research has been mostly confined within the network science community, despite its inherent potential to transcend the conventional boundaries of the PDE-based approach to Turing patterns. Moreover, network topology allows for novel dynamics to be generated via a universal formalism that can be readily extended to account for higher-order structures. The interplay between continuous and discrete settings can pave the way for further developments in the field.

Figures (8)

• Figure 1: Panel (a) depicts the reaction-diffusion network setting: the two species $u$ and $v$ interact within the nodes, which are well-stirred, while diffusing through the links. In panels (b-c) we visualize the conditions for the onset of the Turing instability: on networks, these are only necessary but not sufficient. In fact, the continuous dispersion relation can always be positive (blue line), while the discrete version (red circles) can be either positive (b) or negative(c).
• Figure 2: Turing patterns for the Brusselator model PrigogineNicolis1967 on a Small-World network watts1998collective of $100$ nodes. In panel (a), the time evolution of the concentration of the activator $u$ on each node is shown (qualitatively analogous results are obtained for the inhibitor $v$). The perturbation superimposed to the homogeneous equilibrium triggers the system unstable and the successive evolution makes an inhomogeneous state to spontaneously emerge. In panel (b), the final pattern, again for the concentration of species $u$ on each node, is displayed. The nodes are ordered as in the $1$D lattice from which the Small-World network is generated. It results into the segregation of nodes that are rich/poor of the activator $u$ (similarly for $v$). Finally, panel (c) depicts the concentration of the activator $u$ on the network nodes.
• Figure 3: Panel (a) shows the instability region in the complex plane. When condition \ref{['eq:instregion']} is satisfied, the spectrum of the Laplacian (red circles) reaches the instability region (cyan region). Let us observe that in this case the instability region does not intersect the negative real axis and, thus, a symmetric network would not achieve instability, the spectrum being real. The same setting is represented through the real dispersion relation, shown in panel (b). Here, we can observe that the effect of the non-zero imaginary part of the Laplacian spectrum consists in a detachment from the symmetric case (blue curve), allowing the dispersion relation (red circles) to reach the instability region. The inset of panel (b) shows an example of an oscillatory Turing pattern for the Brusselator model PrigogineNicolis1967 on a directed Newman-Watts network newman1999scaling of $100$ nodes. The oscillations are possible because of the directed topology, as explained in the text. Inspired by Fig. 1 of Asllani1.
• Figure 4: In panel (a), we show the effect of non-normality on a linear dynamics, namely on the $2\times 2$ linear system $\dot{\vec{x}}=A_i\vec{x}$, with $i={1,2}$. Both cases involve stable matrices, $A_1$ and $A_2$, and the two solutions relax to the equilibrium after being perturbed. Nonetheless, $A_1$ is such that its numerical abscissa is positive, i.e., $\omega(A_1)>0$. Then, the system undergoes a transient growth (red curve), at variance with the evolution observed for $A_2$, whose numerical abscissa is negative (green curve). When the dynamics is nonlinear, such effect can dramatically change the fate of the system and, in panel (b), we depict a case study relevant to Turing mechanism. Here, the linear stability analysis would predict no patterns for both systems, as both cases (symmetric - green circles, directed - red circles) return negative values for the dispersion relation. The directed network is, however, non-normal and the transient dynamics caused by non-normality allows the system to "escape" from the basin of attraction of the homogeneous solution, yielding asymptotic Turing patterns (shown in the network). Adapted from Figs. 2 and 4 of jtb, where the model is the Brusselator PrigogineNicolis1967; reproduced with permission.
• Figure 5: Panel (a), illustrative example of a layer-homogeneous fixed point as obtained for the case of a multiplex network composed of five adjacent Watt-Strogatz layers. Here the Brusselator model is assumed to govern the reactive dynamics. Remarkably enough, the system spontaneously evolves towards an asymptotic state where the depicted species displays a different density (as exemplified by the color associated to individual nodes) depending on the layer it belongs to. Panel (b) reports on the emergence of localized patterns for the Brusselator model evolving on a multi-graph. More specifically, we assume that species diffuse on a multigraph network, namely the activator has access to the links encoded by the adjacency matrix $A_0$, while the inhibitor moves by using the links associated to the network described by the matrix $A(\epsilon) = A_0 + \epsilon(A_1 - A_0)$, where $\epsilon \in [0,1]$ gauges the relative importance of the two supports. Stated differently, we can tune the channels available to the inhibitor to coincide with those used by the activator by setting $\epsilon=0$, or to allow for a completely different path of movement if $\epsilon=1$. The ensuing patterns can be modulated, from localized to periodic depending on the choice of $\epsilon$. The obtained patterns (color intensity displayed at the node level) are anticipated by looking at the characteristics of the most unstable eigenvector as revealed by a linear stability analysis. For the three inspected examples, the relevant eigenvector is plotted with an apposite color code. It also encloses the network setting to which it refers to (plot surrounding) to facilitate the comparison with the ensuing dynamical pattern. Panel (a) is Fig. 1 of busiello_homogLturing, while panel (b) is Fig.4 of Asllani2016; reproduced with permission.