Mackey-Glass

The Mackey-Glass equation x' = beta x(t - tau) / (1 + x(t - tau)^n) - gamma x of blood cell production.

• Name: mackey_glass
• Field: biology/physiology
• Formalism: ordinary differential equations
• References: Mackey and Glass 1977; Erneux 2009

Factory

@register_model(
    field='biology/physiology',
    tags=('chaos', 'delay', 'physiology'),
    references=('Mackey and Glass 1977', 'Erneux 2009'),
    defaults={'until': 500.0, 'dt': 0.5},
)
def mackey_glass(
    x: float = 0.5,
    *,
    beta: float = 0.2,
    gamma: float = 0.1,
    n: float = 10.0,
    tau: float = 17.0,
) -> System:
    """Return the Mackey-Glass equation `x' = beta x(t - tau) / (1 + x(t - tau)^n) - gamma x` of blood cell production.

    It is periodic for `tau = 6` and chaotic for `tau = 17`. The history before the start is the initial value.

    Args:
        x:
            The initial concentration, also its history.

        beta:
            The production rate.

        gamma:
            The decay rate.

        n:
            The steepness of the production feedback.

        tau:
            The delay of production.

    Returns:
        system:
            The model, with the concentration `cells.x`.

    Examples:
        >>> from skcomplex.models.biology.physiology import mackey_glass
        >>> mackey_glass(tau=6.0).elements[0].tau
        6.0
    """
    return System(elements=[MackeyGlassElement('cells', x=x, beta=beta, gamma=gamma, n=n, tau=tau)])

Components

class MackeyGlassElement(Element):
    """Concentration `x` of circulating blood cells, made at a delayed Hill rate and lost at a constant rate.

    The concentration follows `x' = beta x(t - tau) / (1 + x(t - tau)^n) - gamma x`.

    Args:
        label:
            The label of the element.

        visual:
            How the element is drawn, or `None` for `default_visual_`.

        x:
            The initial concentration, also its history.

        beta:
            The production rate.

        gamma:
            The loss rate.

        n:
            The Hill coefficient of the production.

        tau:
            The delay of the production.

    Examples:
        >>> from skcomplex.models.biology.physiology import MackeyGlassElement
        >>> MackeyGlassElement('blood', tau=6.0).tau
        6.0
    """

    def __init__(
        self,
        label: str,
        visual: VisualSpec | None = None,
        *,
        x: Annotated[float, StateSpec(domain=RealSet.closed_open(0, oo))] = 0.5,
        beta: Annotated[float, ParameterSpec()] = 0.2,
        gamma: Annotated[float, ParameterSpec()] = 0.1,
        n: Annotated[float, ParameterSpec()] = 10.0,
        tau: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 17.0,
    ) -> None:
        super().__init__(label, visual=visual)
        self.x = x
        self.beta = beta
        self.gamma = gamma
        self.n = n
        self.tau = tau

    def _get_laws(self, s: SimpleNamespace) -> list[BaseLaw]:
        """Return the delayed production and the loss of the cells.

        Args:
            s:
                The namespace of the symbols the laws can use.

        Returns:
            laws:
                The laws of the component.
        """
        history = delayed(s.x, s.tau)
        return [RateLaw(s.x, s.beta * history * (1 - hill(history, 1, s.n)) - s.gamma * s.x)]

Equations

cells.x: diff(x_cells(t), t) == -beta_cells*(x_cells(t - tau_cells)^n_cells/(x_cells(t - tau_cells)^n_cells + 1) - 1)*x_cells(t - tau_cells) - gamma_cells*x_cells(t)