Effect of diversity distribution symmetry on global oscillations of networks of excitable units

TL;DR

The paper investigates how the symmetry of the diversity distribution among coupled FitzHugh-Nagumo units controls global network oscillations, beyond the traditional focus on the oscillator/excitable unit ratio. Through mean-field analysis and extensive simulations across cubic lattice, all-to-all, and Newman-Watts topologies, it introduces two symmetry metrics, $\text{nCOM}$ and $\text{SBS}$, to quantify distribution symmetry and predict oscillatory outcomes. It demonstrates that symmetric bias distributions can sustain resonant global oscillations even when all units are intrinsically excitable, while asymmetry tends to suppress collective activity; a minimal two-unit effective pseudo-potential provides a mechanistic explanation via a cyclic valley in the landscape. These findings offer a general mechanism for symmetry-driven synchronization in heterogeneous excitable networks, with potential implications for physiological systems such as pancreatic islets and neural assemblies.

Abstract

We investigate the role of the degree of symmetry of the diversity distribution in shaping the collective dynamics of networks of coupled excitable units modeled by FitzHugh-Nagumo equations. While previous studies have focused primarily on the ratio between the numbers of individually oscillatory and excitable units, we show that the symmetry of the diversity distribution plays a fundamental role in the emergence of global network oscillations. By exploring various symmetric and asymmetric distributions and simulating network dynamics across various topologies, we demonstrate that symmetric distributions promote resonant collective oscillations even in the absence of oscillatory units. We propose two quantitative metrics, the normalized center of mass and the symmetry balance score, to assess the degree of symmetry and predict the presence or absence of global oscillations. By studying a minimal two-unit system and its effective pseudo-potential, we show that symmetry enables the formation of a landscape characterized by a cyclic valley supporting limit cycles, whereas asymmetry collapses the system into a single non-oscillatory equilibrium. These results provide a general mechanism by which network symmetry drives emergent synchronization in heterogeneous excitable systems.

Figures (11)

• Figure 1: Nullclines of Eqs. (\ref{['global']}) for different values of diversity $M$. System parameters are: $a=60$, $b=1.45$, $J_\mathrm{av}=0$.
• Figure 2: Truncated half-normal diversity distribution comprising oscillatory units only ($J_\mathrm{av}=0$, $\sigma=0.5$). The yellow area highlights the portion of the Gaussian that has been used to sample $J_i$ values. In this case, the lack of symmetry has no negative impact on global network oscillations. The nCOM parameter (nCOM = 0.50, Table \ref{['tab:asymmetry_summary']}) correctly predicts collective oscillations.
• Figure 3: Truncated half-normal diversity distribution comprising excitable units only. The yellow area highlights the portion of the Gaussian that has been used to sample $J_i$ values. In this case, the lack of symmetry prevents global network oscillations. The nCOM parameter (nCOM = 1.50, Table \ref{['tab:asymmetry_summary']}) correctly predicts the absence of global oscillations.
• Figure 4: Truncated half-normal diversity distribution comprising both oscillatory and excitable units in a $50/50\%$ ratio. The yellow area highlights the portion of the Gaussian that has been used to sample $J_i$ values. In this case, the lack of symmetry has no negative impact on global network oscillations. The nCOM parameter (nCOM = 0.99, Table \ref{['tab:asymmetry_summary']}) correctly predicts this.
• Figure 5: Truncated half-normal diversity distribution comprising both oscillatory and excitable units in a 1/2 oscillatory/excitable ratio. The yellow area highlights the portion of the Gaussian that has been used to sample $J_i$ values. In this case, the degree of distribution asymmetry is such that it determines the absence of sustained global network oscillations. The nCOM parameter (nCOM = 1.52, Table \ref{['tab:asymmetry_summary']}) correctly predicts this.