Polyhomeostatic neuron
A neuron that adapts its gain and threshold so its firing rate approaches an exponential distribution.
- Name:
polyhomeostatic_neuron - Field:
biology/neuroscience - Formalism: ordinary differential equations
- References: Triesch 2005; Markovic and Gros 2010; Gros 2015
Factory
@register_model(
field='biology/neuroscience',
tags=('neuron', 'homeostasis', 'adaptation', 'information'),
references=('Triesch 2005', 'Markovic and Gros 2010', 'Gros 2015'),
defaults={'until': 400.0, 'dt': 1.0},
)
def polyhomeostatic_neuron(
*,
mu: float = 0.2,
eps_a: float = 0.05,
eps_b: float = 0.05,
I0: float = 0.0,
A: float = 1.0,
omega: float = 1.0,
) -> System:
"""Return a neuron that adapts its gain and threshold so its firing rate approaches an exponential distribution.
Polyhomeostatic optimization targets the whole distribution of the firing rate, the one of maximal entropy for a
given mean `mu`, rather than its mean alone. The neuron is driven by a periodic input, and the time average of its
rate approaches `mu` as gain and threshold adapt.
Args:
mu:
The target mean firing rate.
eps_a:
The adaptation rate of the gain.
eps_b:
The adaptation rate of the threshold.
I0:
The mean input.
A:
The amplitude of the periodic input.
omega:
The angular frequency of the periodic input.
Returns:
system:
The model, with the potential `neuron.x`, the gain `neuron.a`, and the threshold `neuron.b`.
Examples:
>>> from skcomplex.models.biology.neuroscience import polyhomeostatic_neuron
>>> polyhomeostatic_neuron(mu=0.1).elements[0].mu
0.1
"""
neuron = PolyhomeostaticNeuronElement('neuron', mu=mu, eps_a=eps_a, eps_b=eps_b, I0=I0, A=A, omega=omega)
interactions = [
RelaxationInteraction(
'membrane',
target='neuron',
state='x',
goal=lambda s: s.target.I0 + s.target.A * sin(s.target.omega * s.t),
time=lambda s: 1 / s.target.Gamma,
),
RateInteraction(
'gain_plasticity',
target='neuron',
state='a',
rate=lambda s: s.target.eps_a * (1 / s.target.a + (s.target.x - s.target.b) * _kl_gradient(s)),
),
RateInteraction(
'threshold_plasticity',
target='neuron',
state='b',
rate=lambda s: -s.target.eps_b * s.target.a * _kl_gradient(s),
),
]
return System(elements=[neuron], interactions=interactions)
Components
class PolyhomeostaticNeuronElement(Element):
"""Rate neuron that adapts its gain and threshold so that its firing rate becomes exponentially distributed.
The membrane potential `x` integrates the input `I0 + A sin(omega t)` with rate `Gamma`, and the neuron fires at
rate `y = 1 / (1 + exp(a (b - x)))`. Stochastic gradient descent on the Kullback-Leibler divergence between the
distribution of `y` and an exponential of mean `mu` gives `a' = eps_a (1/a + (x - b) d)` and `b' = -eps_b a d`
with `d = 1 - (2 + 1/mu) y + y²/mu`. The element holds the states and parameters and has no laws of its own: the
relaxation and plasticity interactions of `polyhomeostatic_neuron` read its parameters.
Args:
label:
The label of the neuron.
visual:
How the element is drawn, or `None` for `default_visual_`.
x:
The initial membrane potential.
a:
The initial gain.
b:
The initial threshold.
mu:
The target mean firing rate.
eps_a:
The adaptation rate of the gain.
eps_b:
The adaptation rate of the threshold.
Gamma:
The leak rate of the membrane potential.
I0:
The mean input.
A:
The amplitude of the periodic input.
omega:
The angular frequency of the periodic input.
Examples:
>>> from skcomplex.models.biology.neuroscience import PolyhomeostaticNeuronElement
>>> PolyhomeostaticNeuronElement('neuron').mu
0.2
"""
def __init__(
self,
label: str,
visual: VisualSpec | None = None,
*,
x: Annotated[float, StateSpec()] = 0.0,
a: Annotated[float, StateSpec(domain=RealSet.open(0, oo))] = 1.0,
b: Annotated[float, StateSpec()] = 0.0,
mu: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 0.2,
eps_a: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.05,
eps_b: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 0.05,
Gamma: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 1.0,
I0: Annotated[float, ParameterSpec()] = 0.0,
A: Annotated[float, ParameterSpec(domain=RealSet.closed_open(0, oo))] = 1.0,
omega: Annotated[float, ParameterSpec(domain=RealSet.open(0, oo))] = 1.0,
) -> None:
super().__init__(label, visual=visual)
self.x = x
self.a = a
self.b = b
self.mu = mu
self.eps_a = eps_a
self.eps_b = eps_b
self.Gamma = Gamma
self.I0 = I0
self.A = A
self.omega = omega
class RelaxationInteraction(RateInteraction):
"""Relaxation `(goal(ns) - x) / time(ns)` of a state `x` of a target towards a goal, with optional noise.
With a constant goal and time and an additive noise of amplitude `sigma`, the state follows the Ornstein-Uhlenbeck
process `dx = (goal - x) / time dt + sigma dW`, whose stationary variance is `sigma**2 time / 2`. Without noise it
is first-order smoothing, or adaptive expectation, towards the goal. A goal or a time function takes the namespace
of the interaction, as for `RateInteraction`, and a number stands for a constant. A goal that is a number applies
to every component of a vector state.
Args:
label:
The label of the interaction, unique among the parts of its system.
target:
The path of the element or group whose state relaxes.
state:
The name of the state of the target that relaxes.
goal:
A function of the namespace that returns the goal, or a number.
time:
A function of the namespace that returns the relaxation time, or a positive number.
noise:
A function of the namespace that returns the amplitude of the noise, a number, or `None` for no noise.
Raises:
ValidationError:
The state is not a name, or a function is neither callable nor a number.
Examples:
>>> from skcomplex.core.elements import CompartmentElement
>>> from skcomplex.core.interactions import RelaxationInteraction
>>> from skcomplex.core.system import System
>>> from skcomplex.engine.runtime import Simulation
>>> smoothing = RelaxationInteraction('smoothing', target='belief', state='n', goal=10.0, time=0.5)
>>> system = System(elements=[CompartmentElement('belief', n=0.0)], interactions=[smoothing])
>>> trajectory = Simulation(system, dt=0.01).check().prepare().run(until=5.0).trajectory_
>>> round(float(trajectory.values('belief.n')[-1]), 3)
10.0
"""
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec()],
state: str = 'x',
goal: Callable[[SimpleNamespace], object] | float = 0.0,
time: Callable[[SimpleNamespace], object] | float = 1.0,
noise: Callable[[SimpleNamespace], object] | float | None = None,
) -> None:
super().__init__(label, target=target, state=state, noise=noise)
self.goal = goal
self.time = time
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."""
super()._check_functions()
for name in ('goal', 'time'):
value = getattr(self, name)
if not (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)
def _get_goal(self, namespace: SimpleNamespace) -> object:
"""Return the goal, the hook a subclass overrides to derive it from its parameters."""
return self._evaluate(self.goal, namespace)
def _get_time(self, namespace: SimpleNamespace) -> object:
"""Return the relaxation time, the hook a subclass overrides to derive it from its parameters."""
return self._evaluate(self.time, namespace)
def _get_rate(self, namespace: SimpleNamespace) -> object:
"""Return the relaxation `(goal - x) / time` of the state.
Args:
namespace:
The namespace of the symbols the rate can use.
Returns:
rate:
The rate of the state.
"""
state = getattr(self._get_member(namespace, 'target'), self.state)
goal = self._broadcast(self._get_goal(namespace), state)
return (goal - state) / self._get_time(namespace)
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.elements import CompartmentElement
>>> from skcomplex.core.interactions import RateInteraction
>>> from skcomplex.core.system import System
>>> from skcomplex.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).check().prepare().run(until=2.0).trajectory_
>>> round(float(trajectory.values('stock.n')[-1]), 3)
2.943
"""
def __init__(
self,
label: str,
*,
target: Annotated[str, RoleSpec()],
state: str = 'x',
rate: Callable[[SimpleNamespace], object] | float = 0.0,
sources: Annotated[Sequence[str], RoleSpec(variadic=True)] = (),
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
neuron.x: diff(x_neuron(t), t) == (A_neuron*sin(omega_neuron*t) + I0_neuron - x_neuron(t))*Gamma_neuron
neuron.a: diff(a_neuron(t), t) == (((1/mu_neuron + 2)/(e^((b_neuron(t) - x_neuron(t))*a_neuron(t)) + 1) - 1/(mu_neuron*(e^((b_neuron(t) - x_neuron(t))*a_neuron(t)) + 1)^2) - 1)*(b_neuron(t) - x_neuron(t)) + 1/a_neuron(t))*eps_a_neuron
neuron.b: diff(b_neuron(t), t) == eps_b_neuron*((1/mu_neuron + 2)/(e^((b_neuron(t) - x_neuron(t))*a_neuron(t)) + 1) - 1/(mu_neuron*(e^((b_neuron(t) - x_neuron(t))*a_neuron(t)) + 1)^2) - 1)*a_neuron(t)