Exploring the Interplay of Excitatory and Inhibitory Interactions in the Kuramoto Model on Circle Topologies

Abstract

In the field of collective dynamics, the Kuramoto model serves as a benchmark for the investigation of synchronization phenomena. While mean-field approaches and complex networks have been widely studied, the simple topology of a circle is still relatively unexplored, especially in the context of excitatory and inhibitory interactions. In this work, we focus on the dynamics of the Kuramoto model on a circle with positive and negative connections paying attention to the existence of new attractors different from the synchronized state. Using analytical and computational methods, we find that even for identical oscillators, the introduction of inhibitory interactions significantly modifies the structure of the attractors of the system. Our results extend the current understanding of synchronization in simple topologies and open new avenues for the study of collective dynamics in physical systems.

Figures (10)

• Figure 1: The model of oscillators in a ring geometry with bidirectional couplings.
• Figure 2: Contribution of a single solution of type b) when all links are positive
• Figure 3: Contribution of two consecutive solutions of type b) when all links are positive
• Figure 4: The probability of the attractor characterized by the lag number $\ell$, $P(\ell)$, is reported on the vertical axis. For a ring of $N=100$ nodes and $n$ negative links we computed $6 \cdot 10^4$ realizations. For the sake of clarity, we neglect bins with less than 10 data points. For $n$ even, the only accessible attractors are the ones with $\ell$ even, and their frequency is independent of $n$ (except statistical fluctuations). Conversely, $n$ odd corresponds to $\ell$ odd and $P(\ell)$ is independent of $n$ as well. For all configurations besides $\ell=0$ we have two almost overlapping points because, due to the symmetries of the system, $P(\ell)\sim P(-\ell)$.
• Figure 5: Second moment of the lag number $\ell$ distribution as a function of $N$. For small networks, the symmetries constraint the available values for $\left\langle \ell^{2}\right\rangle$ in case of odd or even number of negative edges. A linear fit for $N\ge10$ shows $\kappa = 0.12807 \pm 0.0001$, with intercept compatible with zero.