Szabó cell system
The cells of Szabó et al. (2006) in a periodic box, which move together above a critical density.
- Name:
szabo_cells - Field:
physics/active_matter - Formalism: stochastic differential equations
- References: Szabó et al. 2006; Vicsek and Zafeiris 2012
Select this system
from skcomplex.systems import create_simulation, create_system, describe
system = create_system('szabo_cells')
entry = describe('szabo_cells')
trajectory = create_simulation('szabo_cells').prepare().run().trajectory_
Factory
@register_system(
field='physics/active_matter',
tags=('active matter', 'sde', 'flocking', 'tissue', 'neighbours'),
references=('Szabó et al. 2006', 'Vicsek and Zafeiris 2012'),
defaults={'until': 60.0, 'dt': 0.5, 'solver': EulerMaruyamaSolver(dt=0.01), 'backend': NumbaBackend()},
observables=lambda system: [
ObservableSpec('mean_heading', _heading, group='cells'),
ObservableSpec('polarization', _polarization),
],
)
def szabo_cells(
n_cells: int = 200,
*,
density: float = 1.0,
eta: float = 0.6,
v0: float = 1.0,
mu: float = 1.0,
tau: float = 1.0,
R_eq: float = 5.0 / 6.0,
R_0: float = 1.0,
F_rep: float = 30.0,
F_adh: float = 0.75,
random_state: int | np.random.Generator | None = 0,
) -> System:
"""Return the cells of Szabó et al. (2006) in a periodic box, which move together above a critical density.
A cell moves at `x' = v0 n + mu F`, where `F` sums the pair forces of the neighbours within `R_0`, the diameter of a
cell, and its heading turns towards its velocity, `theta' = arcsin((n x v / |v|) . e_z) / tau + xi`. No cell reads
the headings of the others, and adhesion alone orders them. With `eta = 0.6` the transition lies at the number
density `rho_c = 0.18 / R_0^2`.
Args:
n_cells:
The number of cells.
density:
The number of cells per unit area.
eta:
The strength of the angular noise.
v0:
The speed of the motility.
mu:
The mobility.
tau:
The relaxation time of the heading.
R_eq:
The equilibrium distance of a pair.
R_0:
The distance beyond which a pair exerts no force.
F_rep:
The largest repulsive force.
F_adh:
The largest adhesive force.
random_state:
The seed or generator of the initial positions and headings.
Returns:
system:
The system, with the positions `cells.x`, the headings `cells.theta`, and the observables `mean_heading` and
`polarization`.
Examples:
>>> from skcomplex.systems.physics.active_matter import szabo_cells
>>> szabo_cells(16, density=0.5).space.extent
(5.656854249492381, 5.656854249492381)
"""
size = math.sqrt(n_cells / density)
rng = np.random.default_rng(random_state)
cells = GroupElement(
'cells',
element=MotileCellElement,
size=n_cells,
values={
'x': rng.uniform(0.0, size, (n_cells, 2)),
'theta': rng.uniform(0.0, 2 * math.pi, n_cells),
'v0': v0,
'mu': mu,
'tau': tau,
'eta': eta,
'R_eq': R_eq,
'R_0': R_0,
'F_rep': F_rep,
'F_adh': F_adh,
},
visual=VisualSpec('points', {'position': 'x', 'radius': R_0 / 2, 'color_by': 'theta'}),
)
motion = RateInteraction('motion', target='cells', state='x', rate=functools.partial(_velocity, reach=R_0))
turning = RateInteraction(
'turning',
target='cells',
state='theta',
rate=functools.partial(_turning, reach=R_0),
noise=_angular_noise,
)
return System(
elements=[cells],
interactions=[motion, turning],
space=ContinuousSpace(extent=(size, size), boundary='periodic'),
)
Components
Elements:
MotileCellElement, fromElement
Interactions:
RateInteraction
class MotileCellElement(Element):
"""Motile cell of a tissue, with its position, the heading of its motility, and its force and motility parameters.
The position follows `x' = v0 n + mu F`, where `n` is the unit vector of the heading and `F` the sum of the pair
forces. A pair force is `F_rep (d - R_eq) / R_eq` towards the neighbour below the distance `R_eq`, and
`F_adh (d - R_eq) / (R_0 - R_eq)` towards it up to the distance `R_0`, the cell diameter.
Args:
label:
The label of the cell or the group.
visual:
How the element is drawn, or `None` for `default_visual_`.
x:
The initial position.
theta:
The initial heading.
v0:
The speed of the motility.
mu:
The mobility.
tau:
The relaxation time of the heading.
eta:
The strength of the angular noise, whose variance per unit time is `eta^2 / 12`.
R_eq:
The equilibrium distance of a pair.
R_0:
The distance beyond which a pair exerts no force.
F_rep:
The largest repulsive force.
F_adh:
The largest adhesive force.
Examples:
>>> from skcomplex.systems.physics.active_matter import MotileCellElement
>>> MotileCellElement('cell').F_rep
30.0
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
x: Annotated[npt.ArrayLike, StateSpec(vector=True, role='position')] = 0.0,
theta: Annotated[float, StateSpec(period=2 * pi, role='direction')] = 0.0,
v0: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 1.0,
mu: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 1.0,
tau: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 1.0,
eta: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.6,
R_eq: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 5.0 / 6.0,
R_0: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 1.0,
F_rep: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 30.0,
F_adh: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.75,
) -> None:
super().__init__(label, visual=visual)
self.x = x
self.theta = theta
self.v0 = v0
self.mu = mu
self.tau = tau
self.eta = eta
self.R_eq = R_eq
self.R_0 = R_0
self.F_rep = F_rep
self.F_adh = F_adh
class RateInteraction(Interaction):
"""Rate `rate(ns)` and noise `noise(ns) dW` added to the highest time derivative of a state of a target.
On a state of order 1 the rate is its first derivative, and on a state of order `k` it adds to the `k`-th
derivative, so rates on one state superpose. With a divisor `d` the rate and the noise are divided by `d`, the name
of a state or parameter of the target, such as an inertia, a friction, or a capacitance, which backends can read.
A rate, a noise, or a divisor function takes the namespace of the interaction, with the namespaces `target` and
`sources` of the bound elements, the time `t`, and the symbols of the parameters a subclass declares. A number
stands for a constant, and a number on a vector state applies to every component. The noise is a Wiener process per
component and member by default, a process of a `WienerProcessSpec` to share or correlate it across states and
members, or a fractional Brownian motion of a Hurst index.
Args:
label:
The label of the interaction, unique among the parts of its system.
target:
The path of the element or group whose state the rate changes.
state:
The name of the state of the target that the rate changes.
rate:
A function of the namespace that returns the rate, or a number.
sources:
The paths of the elements or groups the rate and the noise read but do not change.
noise:
A function of the namespace that returns the amplitude of the noise, a number, or `None` for no noise.
divisor:
The name of the state or parameter of the target that divides the rate and the noise, a function of the
namespace of the target that returns the divisor, such as a moment of inertia `m l^2`, or `None`.
process:
The Wiener processes that drive the noise, or `None` for one independent process per component and member.
index:
The process of `process` that drives the noise, or `None` for every process, one per component of a vector
state.
convention:
The stochastic calculus of the noise, `'ito'`, `'stratonovich'`, or `'anti-ito'`.
hurst:
The Hurst index in `(0, 1)` of a fractional Brownian motion that drives the noise, a number or a function
of the namespace that returns a parameter, or `None` for a Wiener process.
conserved:
Whether the noise is the divergence of a random flux on the links of the lattice of a field, so it keeps the
total of the state.
Raises:
ValidationError:
The state is not a name, a function is neither callable nor a number, the convention is unknown, the Hurst
index is outside `(0, 1)`, or the noise options cannot combine.
Examples:
>>> from skcomplex.core.components.elements import CompartmentElement
>>> from skcomplex.core.components.interactions import RateInteraction
>>> from skcomplex.core.components.system import System
>>> from skcomplex.core.engine.runtime import Simulation
>>> decay = RateInteraction('decay', target='stock', state='n', rate=lambda s: -0.5 * s.target.n)
>>> system = System(elements=[CompartmentElement('stock', n=8.0)], interactions=[decay])
>>> trajectory = Simulation(system, dt=0.01).prepare().run(until=2.0).trajectory_
>>> round(float(trajectory.values('stock.n')[-1]), 3)
2.943
"""
law_classes: ClassVar[tuple[type[BaseLaw], ...]] = (RateLaw, NoiseLaw, FractionalNoiseLaw)
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec(elements=(ElementKind.ANY,), states=('state', 'divisor'))],
state: str = 'x',
rate: Callable[[SimpleNamespace], object] | float = 0.0,
sources: Annotated[Sequence[str], RoleSpec(variadic=True, elements=(ElementKind.ANY,))] = (),
noise: Callable[[SimpleNamespace], object] | float | None = None,
divisor: str | Callable[[SimpleNamespace], object] | None = None,
process: Annotated[WienerProcessSpec | None, RoleSpec(noise=True)] = None,
index: int | None = None,
convention: str = 'ito',
hurst: Callable[[SimpleNamespace], object] | float | None = None,
conserved: bool = False,
) -> None:
super().__init__(label)
self.target = target
self.state = state
self.rate = rate
self.sources = sources
self.noise = noise
self.divisor = divisor
self.process = process
self.index = index
self.convention = convention
self.hurst = hurst
self.conserved = conserved
def _check_functions(self) -> None:
"""Raise an error for a state that is not a name, or a function that is neither callable nor a number."""
if not isinstance(self.state, str) or not self.state.isidentifier():
error_msg = (
f'{type(self).__name__} {self.label!r} changes the state {self.state!r}, which is not a name of a '
'state.'
)
raise ValidationError(error_msg)
for name in ('rate', 'noise'):
value = getattr(self, name)
if not (value is None or callable(value) or isinstance(value, int | float)):
error_msg = f'The {name} of {self.label!r} is {value!r}. Give a function of the namespace or a number.'
raise ValidationError(error_msg)
if not (
self.divisor is None
or callable(self.divisor)
or (isinstance(self.divisor, str) and self.divisor.isidentifier())
):
error_msg = (
f'The divisor of {self.label!r} is {self.divisor!r}. Give the name of a state or parameter, or a '
'function of the namespace of the target.'
)
raise ValidationError(error_msg)
def _check_noise(self) -> None:
"""Raise an error for an unknown convention, a Hurst index outside `(0, 1)`, or noise options that conflict."""
if self.convention not in CONVENTIONS:
error_msg = f'Unknown noise convention {self.convention!r}. Use one of {", ".join(CONVENTIONS)}.'
raise ValidationError(error_msg)
if not (
self.hurst is None or callable(self.hurst) or (isinstance(self.hurst, int | float) and 0 < self.hurst < 1)
):
error_msg = f'The Hurst index of {self.label!r} is {self.hurst!r}. Give a number in (0, 1) or a function.'
raise ValidationError(error_msg)
if self.index is not None and self.process is None:
error_msg = f'{self.label!r} picks the process {self.index!r} but binds no processes. Give `process`.'
raise ValidationError(error_msg)
options = [
name
for name, given in (('hurst', self.hurst), ('process', self.process), ('conserved', self.conserved))
if given
]
if len(options) > 1:
error_msg = f'The noise of {self.label!r} takes {" and ".join(options)}. Give at most one of them.'
raise ValidationError(error_msg)
def _check(self) -> None:
"""Raise an error for a state, functions, or noise options the interaction cannot take.
Raises:
ValidationError:
The type declares states, the state is not a name, a function is neither callable nor a number, the
convention is unknown, the Hurst index is outside `(0, 1)`, or the noise options cannot combine.
"""
super()._check()
self._check_functions()
self._check_noise()
def _evaluate(self, function: Callable[[SimpleNamespace], object] | float, namespace: SimpleNamespace) -> object:
"""Return the value of a function of the namespace, or a number as it is."""
return function(namespace) if callable(function) else function
def _broadcast(self, value: object, state: object) -> object:
"""Return a value with one component per component of a vector state, repeating a number."""
if isinstance(state, FreeModuleElement) and not isinstance(value, FreeModuleElement | list | tuple):
return vector(SR, [value] * len(state))
if isinstance(value, list | tuple):
return vector(SR, value)
return value
def _divide(self, value: object, namespace: SimpleNamespace) -> object:
"""Return a value divided by the divisor of the target, or the value without a divisor."""
if self.divisor is None:
return value
divisor = (
self.divisor(self._get_member(namespace, 'target'))
if callable(self.divisor)
else getattr(self._get_member(namespace, 'target'), self.divisor)
)
return operator.truediv(value, divisor)
def _get_rate(self, namespace: SimpleNamespace) -> object:
"""Return the rate before the divisor, the hook a subclass overrides to derive it from its parameters."""
return self._evaluate(self.rate, namespace)
def _get_noise(self, namespace: SimpleNamespace) -> object:
"""Return the noise amplitude before the divisor, the hook a subclass overrides, or `None` for no noise."""
return None if self.noise is None else self._evaluate(self.noise, namespace)
def _get_noise_law(self, state: object, amplitude: object, namespace: SimpleNamespace) -> BasePart:
"""Return the noise law of an amplitude, by the process, the convention, and the Hurst index."""
if self.hurst is not None:
hurst = self._evaluate(self.hurst, namespace)
return FractionalNoiseLaw(state, amplitude, hurst=hurst, convention=self.convention)
process = None
if self.process is not None:
handle = self._get_member(namespace, 'process').dW
process = handle if self.index is None else handle[self.index]
return NoiseLaw(state, amplitude, convention=self.convention, process=process, conserved=self.conserved)
def _get_laws(self, namespace: SimpleNamespace) -> list[BasePart]:
"""Return the rate law of the state and its noise law, both divided by the divisor.
Args:
namespace:
The namespace of the symbols the laws can use, with the namespaces of the target and the sources.
Returns:
laws:
The rate law, and the noise law when the interaction has noise.
"""
state = getattr(self._get_member(namespace, 'target'), self.state)
rate = self._divide(self._broadcast(self._get_rate(namespace), state), namespace)
laws: list[BasePart] = [RateLaw(state, rate)]
noise = self._get_noise(namespace)
if noise is not None:
laws.append(self._get_noise_law(state, self._divide(noise, namespace), namespace))
return laws
Equations
cells.x[i, 0]: diff(x_cells_0(t), t) == _neighbours_0*mu_cells + v0_cells*cos(theta_cells(t))
cells.x[i, 1]: diff(x_cells_1(t), t) == _neighbours_1*mu_cells + v0_cells*sin(theta_cells(t))
cells.theta[i]: diff(theta_cells(t), t) == arcsin(where(((_neighbours_1*mu_cells + v0_cells*sin(theta_cells(t)))*cos(theta_cells(t)) - (_neighbours_0*mu_cells + v0_cells*cos(theta_cells(t)))*sin(theta_cells(t)))/sqrt((_neighbours_0*mu_cells + v0_cells*cos(theta_cells(t)))^2 + (_neighbours_1*mu_cells + v0_cells*sin(theta_cells(t)))^2 + 1e-12) < -1, -1, where(((_neighbours_1*mu_cells + v0_cells*sin(theta_cells(t)))*cos(theta_cells(t)) - (_neighbours_0*mu_cells + v0_cells*cos(theta_cells(t)))*sin(theta_cells(t)))/sqrt((_neighbours_0*mu_cells + v0_cells*cos(theta_cells(t)))^2 + (_neighbours_1*mu_cells + v0_cells*sin(theta_cells(t)))^2 + 1e-12) > 1, 1, ((_neighbours_1*mu_cells + v0_cells*sin(theta_cells(t)))*cos(theta_cells(t)) - (_neighbours_0*mu_cells + v0_cells*cos(theta_cells(t)))*sin(theta_cells(t)))/sqrt((_neighbours_0*mu_cells + v0_cells*cos(theta_cells(t)))^2 + (_neighbours_1*mu_cells + v0_cells*sin(theta_cells(t)))^2 + 1e-12))))/tau_cells