Synchronization is a ubiquitous phenomenon in physical and biological systems, crucial for understanding phenomena ranging from physiological and brain rhythms to collective animal behavior and coupled Josephson junctions, lasers, and ultracold atoms. While topology's role in theoretical physics, statistical mechanics, and condensed matter is well-established, its influence on synchronization is only beginning to be understood. Recent interest in higher-order networks and simplicial complexes has spurred the study of topology and topological signals in network theory, dynamical systems, and machine learning. Topological signals are dynamical variables associated with nodes and links in a network. While node dynamics are commonly studied, the dynamics of link signals (edge signals) is less explored, despite their relevance to various systems, including brain research (where they can represent synaptic oscillatory signals related to calcium dynamics and synaptic communication), biological transportation networks, and power grids. Previous work on higher-order Kuramoto models showed that coupling the irrotational and solenoidal components of topological signals can lead to abrupt synchronization transitions. However, these models used global adaptive coupling, which is not always realistic, and did not exhibit a stable hysteresis loop on fully connected networks. This paper addresses the open theoretical question of whether a discontinuous transition and a stable hysteresis loop can be achieved with local topological coupling of node and link signals.

The Kuramoto model, a standard model for synchronization, describes the coupling of phases associated with network nodes. It typically exhibits a continuous phase transition. Higher-order Kuramoto models extend this to higher-dimensional simplices of a simplicial complex, using boundary operators to couple the irrotational and solenoidal components of the signal. Global adaptive coupling in these higher-order models can result in discontinuous transitions; however, the global nature of the coupling is not always realistic. Models with constant or space-dependent phase lags have been used to describe various synchronization phenomena, including chimera states and cortical oscillations. The existing models, while insightful, often lack the local topological coupling that is essential for modeling many real-world systems. The present study aims to bridge this gap by introducing a novel model with local adaptive coupling.

This research proposes a new dynamical model called "Dirac synchronization." This model uses the topological Dirac operator and higher-order Laplacians to locally couple the dynamics of topological signals defined on the nodes and links of a network. The model introduces adaptive phase lags for both node and link phases. The coupling mechanism is dictated by the network's topology. The node dynamics is similar to the standard Kuramoto model, but with a time-dependent phase lag depending on nearby link dynamics, and vice-versa. This adaptive local coupling creates a feedback mechanism between node and link signals. The researchers investigated the phase diagram of Dirac synchronization, both numerically and analytically, for a fully connected network. The intrinsic frequencies of nodes and links were drawn from normal distributions. The dynamical equations for node and link phases were solved numerically using a 4th-order Runge-Kutta method. The analytical investigation involved using a continuity equation approach, extending the Ott-Antonsen approach to two dimensions, to study the density distribution of node phases. The stability of the incoherent phase was analyzed to predict the critical coupling constant for the discontinuous forward transition. The rhythmic phase, where the order parameters oscillate, was analyzed using an approximate theoretical framework that classified nodes into four categories based on their phase behavior: frozen, α-oscillating, β-oscillating, and drifting. Finite-size effects on the transition threshold were also studied.

Dirac synchronization exhibits a rich phenomenology distinct from higher-order Kuramoto models. The order parameters are linear combinations of node and link signals, reflecting the interdependent dynamics. On a fully connected network, Dirac synchronization displays an explosive transition with a hysteresis loop. The forward transition is discontinuous, while the backward transition is continuous. The critical coupling constant for the discontinuous forward transition was analytically predicted and matched well with numerical simulations. A rhythmic phase, characterized by non-stationary order parameters, was observed. This rhythmic phase persists over a wide range of coupling strengths and is the only coherent phase observed in the infinite-network limit. The analytical predictions of the critical coupling constants for both forward and backward transitions were found to be in good agreement with extensive numerical simulations. Finite-size effects were analyzed, showing that the discontinuous transition occurs at a coupling strength slightly below the theoretical estimate due to finite-size fluctuations. The study also numerically investigated the rhythmic phase, showing the non-stationary nature of the order parameters and their behavior near the transition between the stationary and rhythmic phases. The angular frequency of oscillations in the rhythmic phase was found to depend on both network size and coupling strength, decreasing with increasing network size and coupling strength. A theoretical framework for analyzing this rhythmic phase was developed, classifying nodes based on their phase trajectories. This classification yielded theoretical expressions for the order parameters that agree well with the numerical results.

The findings demonstrate that Dirac synchronization provides a novel topological mechanism for inducing abrupt, discontinuous synchronization transitions. This mechanism differs from previously proposed frameworks and is driven by the instability of the incoherent phase, leading to a hysteresis loop with a discontinuous forward transition and a continuous backward transition. The emergence of the rhythmic phase, characterized by non-stationary order parameters, offers a potential explanation for the generation of brain rhythms and cortical oscillations. The model's local coupling mechanism makes it biologically plausible for modeling brain dynamics, where global coupling is less realistic. The agreement between analytical predictions and numerical results validates the model and its theoretical underpinnings. The non-stationary coherent phase, or rhythmic phase, is particularly interesting and its characteristics might shed light on the complex dynamics of neuronal populations in the brain.

This paper introduces Dirac synchronization, a novel model for explosive synchronization that leverages the topological Dirac operator to locally couple topological signals on nodes and links of a network. The model exhibits an explosive transition with a hysteresis loop and a rhythmic phase with non-stationary order parameters. These findings provide a new understanding of abrupt synchronization transitions and suggest a potential mechanism for brain rhythms. Future research could extend this model to more complex network topologies and higher-order simplices in simplicial complexes, potentially offering further insights into synchronization phenomena in various real-world systems.

The study primarily focuses on fully connected networks. While this allows for a thorough analytical treatment, the results may not fully generalize to all network topologies. The analysis of the rhythmic phase relies on an approximate theoretical framework, which might require further refinement. The finite-size effects, while considered, could be further investigated to fully account for the deviations from the theoretical predictions observed in finite-size networks.