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Generic balanced synchrony patterns in network dynamics

Romain Joly

TL;DR

The paper addresses the problem of characterizing synchrony patterns in coupled cell networks with symmetry constraints. It employs Henry’s Sard–Smale transversality framework to prove that, for a generic admissible vector field $f\,\in\,\mathcal{C}^1_\ ext{G}$, the synchrony patterns observed along any solution are balanced, i.e., governed by the network’s geometry rather than incidental symmetries. This yields rigorous confirmations of conjectures on rigid synchrony, full oscillation, and constant-state propagation in networks with types, and provides a robust perturbation-based approach (the black-box) to establish genericity under symmetry constraints. The results have implications for interpreting observed synchrony in neural and other networked systems, offering a principled link between trajectory symmetries and the underlying network structure.

Abstract

Coupled cell networks are specific ordinary differential equations with symmetry constraints, which are described by a given directed graph, with cells and arrows divided into several types. The generated dynamics can model, for example, those of neural networks. This type of systems and their emerging symmetries has been the subject of intense study, particularly by Golubitsky, Stewart and their co-authors. In the present article, we show that, for a generic vector field $f$, the synchrony patterns of the solutions of $\dot x(t)=f(x(t))$ are always balanced. This roughly means that the symmetries observed in a solution, such as synchronisation in two different cells, must come from the symmetries imposed by the geometry of network. By doing so, we are completing the proof of several conjectures stated in previous works, including the rigid synchrony conjecture, the full oscillation conjecture and the observation of constant states

Generic balanced synchrony patterns in network dynamics

TL;DR

The paper addresses the problem of characterizing synchrony patterns in coupled cell networks with symmetry constraints. It employs Henry’s Sard–Smale transversality framework to prove that, for a generic admissible vector field , the synchrony patterns observed along any solution are balanced, i.e., governed by the network’s geometry rather than incidental symmetries. This yields rigorous confirmations of conjectures on rigid synchrony, full oscillation, and constant-state propagation in networks with types, and provides a robust perturbation-based approach (the black-box) to establish genericity under symmetry constraints. The results have implications for interpreting observed synchrony in neural and other networked systems, offering a principled link between trajectory symmetries and the underlying network structure.

Abstract

Coupled cell networks are specific ordinary differential equations with symmetry constraints, which are described by a given directed graph, with cells and arrows divided into several types. The generated dynamics can model, for example, those of neural networks. This type of systems and their emerging symmetries has been the subject of intense study, particularly by Golubitsky, Stewart and their co-authors. In the present article, we show that, for a generic vector field , the synchrony patterns of the solutions of are always balanced. This roughly means that the symmetries observed in a solution, such as synchronisation in two different cells, must come from the symmetries imposed by the geometry of network. By doing so, we are completing the proof of several conjectures stated in previous works, including the rigid synchrony conjecture, the full oscillation conjecture and the observation of constant states

Paper Structure

This paper contains 18 sections, 15 theorems, 40 equations, 6 figures.

Table of Contents

  1. Introduction
  2. The network dynamics: definitions, notations and main result
  3. The network with types
  4. The associated ODE with symmetries
  5. Colouring and synchrony
  6. Main result
  7. Genericity and transversality tools
  8. Henry's theorem
  9. An adapted black-box
  10. A strategy for constructing perturbations
  11. Generic synchrony of equilibrium points
  12. Generic synchrony of trajectories
  13. Further results and discussions
  14. Rigid patterns of synchrony
  15. The doubled network and the phase-shift synchrony
  16. ...and 3 more sections

Key Result

Proposition 2.9

The space $\mathcal{C}^1_\mathcal{G}$ is a Baire space: a countable intersection of dense open subsets is dense.

Figures (6)

  • Figure 1: The graph $\mathcal{G}$ above has $4$ cells linked with $8$ arrows. There are two types of cells: the cells $1$ and $3$ (the left/green/squared ones) and the cells $2$ and $4$ (the right/red/round ones). There are two types of arrows: the ones from left to right (the blue/solid ones) and the ones from right to left (the magenta/dashed ones). Notice that we include circling arrows inside the cells to remember that the evolution of a state also depends on itself.
  • Figure 2: Several examples of dynamics inside the network of Figure \ref{['fig_exemple1']}. The symmetries of the network allows solutions presenting some symmetries. The purpose of the present article is to show that, generically, the non-expected symmetries cannot appear. In the above figure, the cases A, B, C and G are compatible with the symmetries of the network and can be observed. The cases D, E, F and H are generically not possible.
  • Figure 3: An example of network with types. The graph $\mathcal{G}$ has $10$ cells, divided in $3$ types, and $17$ arrows divided in $5$ types. The types are coded by shapes and colors. The small circling arrows inside each cell is a reminder that an internal arrow $c_i\rightarrow c_i$ is implicitly present, having its own type, see Definition \ref{['defi_vectfield']} below.
  • Figure 4: A simple illustration of Henry's Theorem. For each $\lambda\in\Lambda$, the function $\Phi(\cdot,\lambda)$ maps a two-dimensional manifold $\mathcal{M}$ into a two-dimensional submanifold of $\mathcal{N}=\mathbb{R}^3$. The kernel of $D_x\Phi(x,\lambda)$ is $\{0\}$ and its index is $-1$ because its image is of codimension $1$. Even if $\Phi(\mathcal{M},\lambda)$ may contain a given point $y_*\in\mathbb{R}^3$ for some specific $\lambda$, if the image of $D\Phi$ contains a direction $Z$ not included in the tangent space $T_{y_*}\Phi(\mathcal{M},\lambda)=\operatorname{R}(D_x\Phi(x,\lambda))$, then $y_*\not\in\Phi(\mathcal{M},\lambda)$ for a generic $\lambda$.
  • Figure 5: To construct a suitable perturbation, we focus on a ball $B$ where $t\mapsto (x_c(t),x_T(t))$ is a bijective curve. Then a function $\phi$ is generated by the combination of bump functions $\phi_n$ with disjoint supports. The choice of the amplitudes $z_n$ of the bumps provides an infinite-dimensional freedom. Moreover, the resulting function $\phi=\sum z_n\phi$ reaches its maximum along the curve $t\mapsto (x_c(t),x_T(t))$.
  • ...and 1 more figures

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Proposition 2.9
  • Definition 2.10
  • ...and 20 more