Hamiltonian control to desynchronize Kuramoto oscillators with higher-order interactions

TL;DR

This paper introduces a Hamiltonian control method to desynchronize Kuramoto oscillators with higher-order interactions, addressing a gap in control strategies for complex brain-like networks.

Contribution

It develops a novel Hamiltonian-based feedback control technique specifically designed for higher-order oscillator interactions, extending beyond pairwise coupling methods.

Findings

Effective in preventing synchronization in synthetic networks

Higher-order control is crucial when pairwise coupling is weak

Minimal controlled nodes can fully desynchronize all-to-all hypergraph networks

Abstract

Synchronization is a ubiquitous phenomenon in nature. Although it is necessary for the functioning of many systems, too much synchronization can also be detrimental, e.g., (partially) synchronized brain patterns support high-level cognitive processes and bodily control, but hypersynchronization can lead to epileptic seizures and tremors, as in neurodegenerative conditions such as Parkinson's disease. Consequently, a critical research question is how to develop effective pinning control methods capable to reduce or modulate synchronization as needed. Although such methods exist to control pairwise-coupled oscillators, there are none for higher-order interactions, despite the increasing evidence of their relevant role in brain dynamics. In this work, we fill this gap by proposing a generalized control method designed to desynchronize Kuramoto oscillators connected through higher-order interactions. Our method embeds a higher-order Kuramoto model into a suitable Hamiltonian flow, and builds up on previous work in Hamiltonian control theory to analytically construct a feedback control mechanism. We numerically show that the proposed method effectively prevents synchronization in synthetics and empirical higher-order networks. Although our findings indicate that pairwise contributions in the feedback loop are often sufficient, the higher-order generalization becomes crucial when pairwise coupling is weak. Finally, we explore the minimum number of controlled nodes required to fully desynchronize oscillators coupled via an all-to-all hypergraphs.