The Kuramoto model on a sphere: Explaining its low-dimensional dynamics with group theory and hyperbolic geometry.

We study a system of N identical interacting particles moving on the unit sphere in d-dimensional space. The particles are self-propelled and coupled all to all, and their motion is heavily overdamped. For d=2, the system reduces to the classic Kuramoto model of coupled oscillators; for d=3, it has been proposed to describe the orientation dynamics of swarms of drones or other entities moving about in three-dimensional space.

Dynamics of the Kuramoto-Sakaguchi oscillator network with asymmetric order parameter.

We study the dynamics of a generalized version of the famous Kuramoto-Sakaguchi coupled oscillator model. In the classic version of this system, all oscillators are governed by the same ordinary differential equation (ODE), which depends on the order parameter of the oscillator configuration. The order parameter is the arithmetic mean of the configuration of complex oscillator phases, multiplied by some constant complex coupling factor.

Cluster synchronization in networks of identical oscillators with α-function pulse coupling.

We study a network of N identical leaky integrate-and-fire model neurons coupled by α-function pulses, weighted by a coupling parameter K. Studies of the dynamics of this system have mostly focused on the stability of the fully synchronized and the fully asynchronous splay states, which naturally depends on the sign of K, i.e.

Classification of attractors for systems of identical coupled Kuramoto oscillators.

We present a complete classification of attractors for networks of coupled identical Kuramoto oscillators. In such networks, each oscillator is driven by the same first-order trigonometric function, with coefficients given by symmetric functions of the entire oscillator ensemble. For [Formula: see text] oscillators, there are four possible types of attractors: completely synchronized fixed points or limit cycles, and fixed points or limit cycles where all but one of the oscillators are synchronized.

The asymptotic behavior of the order parameter for the infinite-N Kuramoto model.

The Kuramoto model, first proposed in 1975, consists of a population of sinusoidally coupled oscillators with random natural frequencies. It has served as an idealized model for coupled oscillator systems in physics, chemistry, and biology. This paper addresses a long-standing problem about the infinite-N Kuramoto model, which is to describe the asymptotic behavior of the order parameter for this system.

Structure of long-term average frequencies for Kuramoto oscillator systems.

We study the long-term average frequency as a function of the natural frequency for Kuramoto oscillators with periodic coefficients. Unlike the case for more general periodically forced oscillators, this function is never a "devil's staircase"; it may have plateaus at integer multiples of the forcing frequency, but we prove it is strictly increasing between these plateaus. The proof uses the fact that the flow maps for Kuramoto oscillators extend to Möbius transformations on the complex plane, and that Möbius transformations have particularly simple dynamics that rule out p:q mode locking except in the case of fixed points (q=1).

Rhythm-induced spike-timing patterns characterized by 1D firing maps.

We explore patterns in the spike timing of neurons receiving periodic inputs, with an emphasis on stable characteristics which are realized in both models and in-vitro whole-cell recordings. We report on whole-cell recordings of pyramidal CA1 cells from rat hippocampus and entorhinal cortex and compare this data to model simulations. Cells were injected with a constant current to induce a steady firing rate and then a modest rhythm was added which altered the spike times and their corresponding phases relative to the rhythm.

Identical phase oscillators with global sinusoidal coupling evolve by Mobius group action.

Systems of N identical phase oscillators with global sinusoidal coupling are known to display low-dimensional dynamics. Although this phenomenon was first observed about 20 years ago, its underlying cause has remained a puzzle. Here we expose the structure working behind the scenes of these systems by proving that the governing equations are generated by the action of the Mobius group, a three-parameter subgroup of fractional linear transformations that map the unit disk to itself.

Dynamical phase transitions in periodically driven model neurons.

Transitions between dynamical states in integrate-and-fire neuron models with periodic stimuli result from tangent or discontinuous bifurcations of a return map. We study their characteristic scaling laws and show that discontinuous bifurcations exhibit a kind of phase transition intermediate between continuous and first order. In the model-independent spirit of our analysis we show that a six-dimensional (6D) gating variable model with an attracting limit cycle has similar phase transitions, governed by a 1D return map.