Kuramoto

The Kuramoto update rule implements a phase-coupled oscillator model inspired by the Kuramoto model of synchronization. Each neuron’s activation is treated as a phase angle \(\theta\), which evolves over time based on the phase differences between itself and its input neurons.

At each time step, the neuron computes:

\[\theta' = \theta + \Delta t \cdot \left( \omega + \frac{1}{N} \sum_j K_j \sin(\theta_j - \theta) \right)\]

Where:

• \(\theta\) is the neuron’s current phase (activation).
• \(\omega\) is the neuron’s natural frequency.
• \(K_j\) is the strength of synapse \(j\).
• \(\theta_j\) is the phase of presynaptic neuron \(j\).
• \(N\) is the number of input synapses.
• \(\Delta t\) is the simulation time step.

The updated phase is then wrapped into the interval \([0, 2\pi]\).

This rule models synchronization phenomena and can be used to simulate coupled oscillators, rhythmic behavior, and collective entrainment in neural networks.

Parameters

• Natural Frequency: The constant intrinsic rotation speed \(\omega\) of the oscillator.

For all other parameters, see common neuron properties