Clustering and synchronization analysis of Networks of Bistable Systems

TL;DR

This work analyzes networks of diffusively coupled bistable systems, establishing global convergence to equilibria for generic monotone regulatory functions and providing a sufficient coupling-condition for global synchronization based on algebraic connectivity. The authors reformulate the network as a CCW positive-feedback interconnection and leverage Angeli’s monotone-system theory to prove convergence to equilibria, with synchronization emerging when coupling is strong enough relative to the regulators’ Lipschitz constants. They further study a biologically relevant all-to-all, piecewise-linear model to locate equilibria, characterize their stability, and describe clustering configurations as the coupling gain varies. Numerical simulations across multiple topologies validate the theory and illustrate the transition from multistability to synchronized equilibria, as well as the comparison between piecewise-linear and Hill-type regulatory functions. The results provide analytical insight into multistability, synchronization, and clustering in diffusion-coupled gene regulatory networks and related biological systems, with implications for quorum-sensing-like coupling and synthetic circuit design.

Abstract

This paper studies the dynamics of a network of diffusively-coupled bistable systems. Under mild conditions and without requiring smoothness of the vector field, we analyze the network dynamics and show that the solutions converge globally to the set of equilibria for generic monotone (but not necessarily strictly monotone) regulatory functions. Sufficient conditions for global state synchronization are provided. Finally, by adopting a piecewise linear approximation of the vector field, we determine the existence, location and stability of the equilibria as function of the coupling gain. The theoretical results are illustrated with numerical simulations.

Figures (6)

• Figure 1: Phase portrait for \ref{['eq:bistable_compartment']} with regulatory functions $g$ in \ref{['eq:g_function']}. The grey box with orange edges highlights the forward invariant (and attractive) set $\mathcal{B}$. The full black circles represent the two stable equilibria and the white circle represents the unstable saddle equilibrium.
• Figure 2: Positive feedback interconnection scheme representing the mathematical model in \ref{['eq:single_system_network']}.
• Figure 3: In (a), the dotted vertical blue lines correspond to the values $k^{\overline{q}}$ computed analytically. The continuous blue line represents the total number of equilibria in all saturated domains as a function of the coupling parameter $k$. The continuous orange line represents the total number of equilibria in all domains (including non-saturated) as a function of the coupling parameter $k$. The dotted red line represents the minimum coupling gain $k^{\lambda}$ to ensure that only synchronized equilibria exist. In all figures, the minimum values for the blue and orange lines are respectively 2 and 3. The system's parameters used are are $V_1 = V_2 = 1$, $\gamma_{1}=\gamma_{2}=1$ and $\theta=0.45$, $\delta=0.1$.
• Figure 4: Synchronization bounds $k^{\lambda}$ (red line) and $k^s$ (blue line) for increasing values of the Lipschitz constant $\ell_2=1/\delta$ of the piecewise affine activation function $g_1(x_1) = g(x_1)$. Parameters used: $N=5$, $V_1 = V_2 = 1$, $\gamma_{1}=\gamma_{2}=1$ and $\theta=0.45$.
• Figure 5: Hill function $g_s (x)$ with cooperativity degree $n=3$ (orange line) and piece-wise affine approximation $g(x)$ (blue line). Parameters used: $\theta_H = 1.5$, $\delta = 2$, $\theta = \theta_H-\delta/2$.

Theorems & Definitions (25)

• Proposition 2.1
• proof
• Definition 2.1
• Theorem 3.1
• Lemma 3.1
• proof
• Proposition 3.1
• proof
• Theorem 4.1
• proof