Dynamics of oscillator populations globally coupled with distributed phase shifts
Authors: Smirnov L. A. , Pikovsky A. S. 
Published: May 2024

Globally coupled populations of oscillators are exemplary models for synchronization and the emergence of collective modes. In many cases, the nature of the global coupling is predetermined by the setup. Nevertheless, the oscillators can possess distinct properties and intrinsic noise, leading to non-identity of the system elements. The variation in natural frequencies among the oscillators is the primary source of non-identity. This feature has already been extensively studied. Our research focuses on the impact of phase shift coupling disorder. These phase lags naturally occur where the global force must propagate to reach spatially distributed elements of the population.
First, we develop a phase model in the Kuramoto–Sakaguchi form for a group of quadratic integrate-and-fire neurons that are inter-connected by a synaptic current transmitted with different time delays. This occurs when each cell receives input from other cells of the ensemble with an inherent delay. From a mathematical perspective, a global force affects oscillators with different time delays, resulting in a spread of phase lags in the corresponding phase model. Assuming a weak interaction between system units, we transform to slow-varying phases and use the standard time-average procedure.
Secondly, we demonstrate that a distribution of phase shifts in coupling may result from local oscillator properties, particularly when synaptic coupling of neurons incorporates a “low-pass filter” of an incoming, globally averaged synaptic field. For this purpose, we examine the classic θ-neuron model, which is frequently used to analyze the collective dynamics of a Type I neuron population. Here, we posit that neurons interact through chemical synapses, with each corresponding synaptic current forcing on the neuron satisfying the relaxation equation while having an individual, disordered value of the relaxation constant. Under this assumption of weak coupling and through the use of multiple timescale analysis, we obtain the Kuramoto-Sakaguchi model of phase oscillators which exhibit distributed phase lags.
Next, we will consider the characteristics of the phase model. In the thermodynamic limit, the one-particle probability density function characterizes the continuum of phase oscillators. It evolves according to the continuity equation and possesses an exact solution in the Ott–Antonsen ansatz form at each α phase lag value. This manifold is attractive and represents a special ansatz for the expansion of a Poisson kernel in a Fourier series with respect to the phase variable, as demonstrated in other papers. Using this analytical approach, we obtain a low-dimensional depiction of the collective behavior of the system, which indicates that in the equations for macroscopic complex fields, the redefined order parameter Q(t,α) depends solely on the phase shifts α through the initial conditions. However, its dynamics remain independent. Using stability analyses in the linear approximation and reduced equations, we argue that during the dynamics process, the memory of the initial state is lost and Q(t,α)→Q(t). After Q(t) converges, the population dynamics with a distribution of phase shifts reduces to a single dynamical equation for the auxiliary order parameter Q(t), and the original order parameters are connected to it through circular moments of the phase shift distribution g(α).
All theoretical concepts are confirmed through numerical calculations performed directly within the oscillatory population models under consideration.