tvb.SupHopf

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

Supercritical Hopf bifurcation oscillator.

Two-state model in Cartesian coordinates representing the normal form of a supercritical Hopf bifurcation, widely used for modeling oscillatory dynamics near bifurcation points.

Notes

State equations:

\[ \begin{aligned} \frac{dx}{dt} &= (a - x^2 - y^2) x - \omega y + c_{\text{delayed},x} + c_{\text{instant}} \\ \frac{dy}{dt} &= (a - x^2 - y^2) y + \omega x + c_{\text{delayed},y} \end{aligned} \]

where:

  • \(a\): Bifurcation parameter (\(a < 0\): stable fixed point, \(a > 0\): limit cycle)
  • \(\omega\): Angular frequency of oscillation
  • Limit cycle amplitude: \(\sqrt{a}\) for \(a > 0\)

The delayed coupling is 2-dimensional, allowing separate coupling for x and y components.

Attributes

Name Type Description
STATE_NAMES tuple of str State variables: ('x', 'y')
INITIAL_STATE tuple of float Default initial conditions: (0.1, 0.0)
AUXILIARY_NAMES tuple of str No auxiliary variables: ()
COUPLING_INPUTS dict Coupling specification: {'instant': 1, 'delayed': 2}
DEFAULT_PARAMS Bunch Parameters: a=-0.5 (bifurcation), omega=1.0 (frequency)

References

Deco et al. (2017). The dynamics of resting fluctuations in the brain: metastability and its dynamical cortical core. Scientific Reports, 7, 3095.

Methods

Name Description
dynamics Compute supercritical Hopf dynamics.

dynamics

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

Compute supercritical Hopf dynamics.

Parameters

Name Type Description Default
t float Current time required
state jnp.ndarray Current state with shape [2, n_nodes] containing (x, y) required
params Bunch Model parameters (a: bifurcation, omega: frequency) required
coupling Bunch Coupling inputs with attributes: - .instant[0]: local coupling (x only) - .delayed[0]: long-range x-coupling - .delayed[1]: long-range y-coupling required
external Bunch External inputs (currently unused) required

Returns

Name Type Description
derivatives jnp.ndarray State derivatives with shape [2, n_nodes]