tvb.Kuramoto

experimental.network_dynamics.dynamics.tvb.Kuramoto(**kwargs)

Kuramoto phase oscillator model.

Single-state model representing the phase angle of an oscillator, widely used to study synchronization phenomena in networks of oscillatory units.

Notes

The model describes synchronization dynamics in networks of coupled oscillators through phase interactions.

State equation:

\[\frac{d\theta}{dt} = \omega + c_{\text{delayed}} + \sin(c_{\text{instant}} \cdot \theta)\]

where:

• \(\theta\): Phase angle \([0, 2\pi]\)
• \(\omega\): Natural frequency of oscillation
• \(c_{\text{instant}}\): Local coupling (phase-dependent via sinusoidal transformation)
• \(c_{\text{delayed}}\): Long-range delayed coupling (additive)

The local coupling uses a sinusoidal transformation capturing the phase-dependent interaction characteristic of Kuramoto-type coupling.

Attributes

Name	Type	Description
STATE_NAMES	tuple of str	State variable: ('theta',)
INITIAL_STATE	tuple of float	Default initial condition: (0.1,)
AUXILIARY_NAMES	tuple of str	No auxiliary variables: ()
COUPLING_INPUTS	dict	Coupling specification: {'instant': 1, 'delayed': 1}
DEFAULT_PARAMS	Bunch	Natural frequency omega=1.0 (rad/s or Hz depending on units)

References

Kuramoto (1984). Chemical Oscillations, Waves, and Turbulence. Springer-Verlag, Berlin.

Methods

Name	Description
dynamics	Compute Kuramoto dynamics.

dynamics

experimental.network_dynamics.dynamics.tvb.Kuramoto.dynamics(
    t,
    state,
    params,
    coupling,
    external,
)

Compute Kuramoto dynamics.

Parameters

Name	Type	Description	Default
t	float	Current time	required
state	jnp.ndarray	Current state with shape [1, n_nodes] containing theta (phase angle)	required
params	Bunch	Model parameters (omega: natural frequency)	required
coupling	Bunch	Coupling inputs with attributes: - .instant[0]: local coupling (used in sin transform) - .delayed[0]: long-range coupling (additive)	required
external	Bunch	External inputs (currently unused)	required

Returns

Name	Type	Description
derivatives	jnp.ndarray	Phase velocity with shape [1, n_nodes]