Synchrony for weak coupling in the complexified Kuramoto model

Abstract

We present the finite-size Kuramoto model analytically continued from real to complex variables and analyze its collective dynamics. For strong coupling, synchrony appears through locked states that constitute attractors, as for the real-variable system. However, synchrony persists in the form of \textit{complex locked states} for coupling strengths $K$ below the transition $K^{(\text{pl})}$ to classical \textit{phase locking}. Stable complex locked states indicate a locked sub-population of zero mean frequency in the real-variable model and their imaginary parts help identifying which units comprise that sub-population. We uncover a second transition at $K^{\prime }<K^{(\text{pl})}$ below which complex locked states become linearly unstable yet still exist for arbitrarily small coupling strengths.

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• Low Dimensional Dynamics of Globally Coupled Complex Riccati Equations: Exact Firing-rate Equations for Spiking Neurons with Clustered Substructure
• Synchronization in the complexified Kuramoto model
• On the Equivalence of Synchronization Definitions in the Kuramoto Flow: A Unified Approach
• Hopf-Induced Desynchronization