Diffusion-driven instability of topological signals coupled by the Dirac operator

TL;DR

The paper extends Turing pattern theory to topological signals living on nodes and links of networks, coupled by the Dirac operator $\mathcal{D}$ and diffusing through Hodge-Laplacians $\mathcal{L}$. It derives analytic conditions for the onset of diffusion-driven instabilities in two settings: Dirac reaction coupling with Laplacian diffusion, and Dirac cross-diffusion terms (linear or cubic) that mediate diffusion across dimensions. Key findings include that Turing patterns are necessarily distributed across both nodes and edges, and that projected signals $\mathcal{D}\Phi$ inherit the pattern; cross-diffusion can enable instability even when the reaction term does not couple across dimensions. Validation on a benchmark network and on square lattices confirms the theoretical predictions and demonstrates the practical relevance for systems with higher-order interactions and topological signals.

Abstract

The study of reaction-diffusion systems on networks is of paramount relevance for the understanding of nonlinear processes in systems where the topology is intrinsically discrete, such as the brain. Until now reaction-diffusion systems have been studied only when species are defined on the nodes of a network. However, in a number of real systems including, e.g., the brain and the climate, dynamical variables are not only defined on nodes but also on links, faces and higher-dimensional cells of simplicial or cell complexes, leading to topological signals. In this work we study reaction-diffusion processes of topological signals coupled through the Dirac operator. The Dirac operator allows topological signals of different dimension to interact or cross-diffuse as it projects the topological signals defined on simplices or cells of a given dimension to simplices or cells of one dimension up or one dimension down. By focusing on the framework involving nodes and links we establish the conditions for the emergence of Turing patterns and we show that the latter are never localized only on nodes or only on links of the network. Moreover when the topological signals display Turing pattern their projection does as well. We validate the theory hereby developed on a benchmark network model and on square lattices with periodic boundary conditions.

Figures (7)

• Figure 1: We schematically represent the dynamical state of a simplicial complex encoded by the vector $\bm\Phi=(u,b,w)^{\top}$ and the vector $\bm\Psi=\mathcal{D}\bm\Phi=(\hat{u},\hat{v},\hat{w})^{\top}$. In particular we represent topological signals and projected topological signals supported on $0$, $1$ and $2$-simplices respectively in panels a), b), and c). The Dirac operator $\mathcal{D}$ projects the topological signals of each dimension either one dimension up or one dimension down, and leads to projected components defined on nodes $(\hat{u}={\bf B}_1v$, links $\hat{v}={\bf B}_2w+{\bf B}_1^{\top}v$, and triangles $\hat{w}={\bf B}_2^{\top}v$. Here $\hat{u}={\bf B}_1v$ describes the link signals projected on the nodes; ${\bf B}_1^{\top}u$ indicates the irrotational component of $\hat{v}$ and describes the projection of the node signals on the links; ${\bf B}_2w$ indicates the solenoidal component of $\hat{v}$ and describes the projection of the triangle signals on the links; finally ${\bf B}_2^{\top}v$ describes the projection of the link signals on the triangles.
• Figure 2: $a)$ Turing patterns for species defined on nodes and on links described by model \ref{['eq:model']} on a network satisfying the conditions for the existence of an homogeneous equilibrium. In panels $b)$ and $c)$ we depict time series of the two species $u$ and $v$ on the nodes and the links, respectively, while panels $d)$ and $e)$ show the time series of the projection of the two species with the action of the boundary operator $\mathbf{B}_1$. The parameters are $a=\alpha=b=\beta=\gamma=D_0=D_1=1$ and $c=6$. The perturbation defining the initial condition, is $\sim10^{-2}$.
• Figure 3: Turing region in the parameters space $(c,\gamma)$. In the main panel (B), we report the region of parameters for which the Turing instability emerges; having fixed $a=\alpha=b=\beta=D_0=D_1=1$ we show the maximum of the real part of dispersion relation as a function of $c$ and $\gamma$, by using a color code (yellow corresponding to large values, red to small but positive ones and white to negative ones). The black solid curves is given by $c\gamma = \sqrt{4 D_0D_1 a\alpha}+\alpha D_0+aD_1$ (see Eq. \ref{['eq:turing']}). Panels A1), A2) and A3) correspond to the choice $(c,\gamma)=(2,2)$ that lies outside the Turing region; one can observe that the dispersion relation (panel A1) is negative and indeed patterns cannot develop as shown by the node (resp. link) amplitude (panel A2) resp. A3) decaying to $0$. Panels C1, C2 and C3) show similar results but for $(c,\gamma)=(6,2)$ inside the Turing region; the dispersion relation (C1) reaches positive values and the node (resp. link) amplitude stabilizes far from zero (C2, resp. C3).
• Figure 4: We report the distribution of the node (a) and link (b) amplitude of the Turing patterns obtained by numerically simulating $5000$ times system \ref{['eq:model']} with the parameters used in Fig. \ref{['fig:fig1']} and by changing the initial conditions.
• Figure 5: $a)$ Turing patterns for the species on the nodes and on the links described by model \ref{['eq:model2']} on a network satisfying the conditions for the homogeneous equilibrium; $b)$ dispersion relation: in blue we depict the continuous curve, computed by replacing the discrete parameter $b^2_k$ with a continuous variable, while the cyan dots are the actual dispersion relation, where now the onset of the Turing instability is a function of the (real) spectrum of $\mathop{\mathrm{\mathbf{L}}}\nolimits_0$, i.e., computed by using the discrete values of $b_k^2$. In panels $c)$ and $d)$ we depict time series of the two species $u$ and $v$ on the nodes and the links, respectively, while panels $e)$ and $f)$ show the time series of the projection of the two species with the action of the boundary operator $B_1$. The parameters are $a=0.8$, $\alpha=1.3$, $b=1$, $\beta=0.5$, $c=8$, $\gamma=2$, $D_0=0.5$, $D_1=1$, $D_{01}=-1.5$ and $D_{10}=0.4$; the initial perturbation is $\sim10^{-2}$.