Topology in neuroscience
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from __future__ import annotations
import typing
from dataclasses import dataclass, field
from itertools import chain
from functools import partial
import sympy as sp
import numpy as np
from utils import get_orientation_phase_grid
sp.init_printing()
k, x0, y0, phi_rf, theta_rf, sigma_x, sigma_y, sigma, x, y, theta_grating, phi_grating = sp.symbols(r'k x_0 y_0 \phi_{rf} \theta_{rf} \sigma_x \sigma_y \sigma x y \theta_{grating} \phi_{grating}')
defaults = {
k: 6,
sigma: 1,
phi_rf: sp.pi / 2,
phi_grating: sp.pi / 2,
theta_grating: 0,
sigma: 1,
x0: 0, y0: 0
}
sigma_x = sigma_y = sigma
grating_f = sp.cos(k * (x - x0) * sp.cos(theta_grating) + k * (y - y0) * sp.sin(theta_grating) + phi_grating)
receptive_field = 1 / (2 * sp.pi * sigma * sigma) * sp.exp(-(x ** 2 + y ** 2) / (2 * sigma ** 2)) * sp.cos(
k * x * sp.cos(theta_rf) + k * y * sp.sin(theta_rf) + phi_rf)
# p = sp.cosh(k ** 2 * sigma ** 2 * sp.cos(theta)) * sp.exp(k ** 2 * (1 + sp.cos(theta) ** 2) / 2) * sp.cos(
# phi - k * (x0 * sp.cos(theta) + y0 * sp.sin(theta)))
# p = sp.cosh(k ** 2 * sigma ** 2 * sp.cos(theta) * 4) * sp.exp(-4 * k ** 2 * sigma ** 2) * sp.cos(
# phi - k * (x0 * sp.cos(theta) + y0 * sp.sin(theta)))
p = (1 / 2) * sp.exp(-k*k*sigma*sigma) * (
sp.exp(-k*k*sigma*sigma*sp.sin(theta_grating + theta_rf)) * sp.cos(phi_grating + phi_rf + 2 * k / (sigma * sigma) * (
x0 * sp.cos(theta_rf) + y0 * sp.sin(theta_grating)
+ x0 * sp.sin(theta_rf) + y0 * sp.cos(theta_grating)
)) +
sp.exp( k*k*sigma*sigma*sp.sin(theta_grating + theta_rf)) * sp.cos(phi_grating - phi_rf + 2 * k / (sigma * sigma) * (
-x0 * sp.cos(theta_rf) - y0 * sp.sin(theta_rf)
+ x0 * sp.sin(theta_rf) + y0 * sp.cos(theta_grating)
)))
sigma_split = np.arange(0.1, 1, 0.05)
k_split = np.arange(0.2, 6, 0.2)
xy_split = np.arange(-1, 1, 0.05)
phi_split = np.arange(0, 2 * np.pi, np.pi / 100)
theta_split = np.arange(0, np.pi, np.pi / 100)
# The second option is (the distribution function, the function that takes the size and returns the step and the starting point)
Distribution = typing.Union[float, typing.Dict[float, float], typing.Tuple[typing.Callable[[float], float], typing.Callable[[int], typing.Tuple[float, float]]]]
def sample_distribution(distribution: Distribution, size: int = 1) -> np.ndarray:
if isinstance(distribution, float) or isinstance(distribution, int):
return np.array([float(distribution)] * size)
if isinstance(distribution, dict):
return np.random.choice(list(distribution.keys()), size, p=list(distribution.values()))
elif isinstance(distribution, tuple):
step, start = distribution[1](size)
res = [start]
for i in range(size):
res.append(res[-1] + step / distribution[0](res[-1]))
return np.array(res)
else:
raise ValueError(f'Unknown distribution type: {type(distribution)}')
def get_uniform_dist(start: float, stop: float) -> Distribution:
return (lambda _x: 1, lambda size: ((stop - start) / (size - 1 or 1), start))
phi_dist_uni = get_uniform_dist(0, 2 * np.pi)
theta_dist_uni = get_uniform_dist(0, np.pi)
def sigmoid(x):
return 1 / (1 + np.exp(-x))
@dataclass
class Cell:
phi_val: float
theta_val: float
sigma_val: float = defaults[sigma]
x0_val: float = defaults[x0]
y0_val: float = defaults[y0]
k_val: float = defaults[k]
@property
def sympy_func(self) -> sp.Expr:
return receptive_field\
.subs(sigma, self.sigma_val).subs(x0, self.x0_val).subs(y0, self.y0_val)\
.subs(k, self.k_val).subs(phi_rf, self.phi_val).subs(theta_rf, self.theta_val)
def get_tuning_function(self) -> typing.Callable[[np.ndarray, np.ndarray], np.ndarray]:
"""
Get the tuning sympy function as a numpy lambda function of theta and phi.
:return: a function (theta, phi) -> value
"""
return sp.lambdify(
(theta_grating, phi_grating),
p.subs(sigma, self.sigma_val).subs(x0, self.x0_val)\
.subs(y0, self.y0_val).subs(k, self.k_val)\
.subs(phi_rf, self.phi_val).subs(theta_rf, self.theta_val),
'numpy')
def get_value(self, theta_deg: float, phi_deg: float) -> float:
return float(self.get_tuning_function()(theta_deg * np.pi / 180, phi_deg * np.pi / 180))
def get_tuning_plot(self, theta_step_deg: float, phi_step_deg: float) -> np.ndarray:
grid = get_orientation_phase_grid(theta_step_deg, phi_step_deg)
return self.get_tuning_function()(grid[:, :, 0], grid[:, :, 1])
@dataclass
class Grating:
phi_val: float = defaults[phi_grating]
theta_val: float = defaults[theta_grating]
k_val: float = defaults[k]
@property
def sympy_func(self) -> sp.Expr:
return grating_f.subs(k, self.k_val).subs(phi_grating, self.phi_val).subs(theta_grating, self.theta_val)
@dataclass
class Population:
cells: typing.List[Cell] = field(default_factory=list)
@property
def response_func(self) -> typing.Callable[[float, float], np.ndarray]:
"""
Use sp.lambdify and the expression to generate the necessary function.
:return: a function (phi, theta) -> responses
"""
return partial(
sp.lambdify(
(x0, y0, k, sigma, phi_rf, theta_rf, phi_grating, theta_grating, ),
p, 'numpy'),
0, 0, np.array([cell.k_val for cell in self.cells]).reshape((-1, 1)),
np.array([cell.sigma_val for cell in self.cells]).reshape((-1, 1)),
np.array([cell.phi_val for cell in self.cells]).reshape((-1, 1)),
np.array([cell.theta_val for cell in self.cells]).reshape((-1, 1)),
)
@classmethod
def sample(cls, n_orient: int, n_phases: int,
phi_dist: Distribution = phi_dist_uni,
theta_dist: Distribution = theta_dist_uni,
sigma_dist: Distribution = defaults[sigma],
k_val: float = defaults[k],
xy_dist: Distribution = get_uniform_dist(-5, 5)):
return cls(cells=[
Cell(phi_val=phi_val, theta_val=theta_val, sigma_val=sigma_val, x0_val=x0_val * 0, y0_val=y0_val * 0, k_val=k_val)
for phi_val, theta_val, sigma_val, x0_val, y0_val in zip(
list(sample_distribution(phi_dist, n_phases)) * n_orient,
list(chain(*[[x] * n_phases for x in sample_distribution(theta_dist, n_orient)])),
sample_distribution(sigma_dist, n_phases * n_orient),
sample_distribution(xy_dist, n_phases * n_orient),
sample_distribution(xy_dist, n_phases * n_orient))
])
def get_response(self, phi_deg: float, theta_deg: float, coef: float = 4, use_sigmoid: bool = True) -> np.ndarray:
return (sigmoid if use_sigmoid else (lambda x: x))(np.array([cell.get_value(theta_deg, phi_deg) for cell in self.cells]) * coef)
def sample_responses(
self, n: int, noise_sigma: float = 0, coef: float = 2, use_sigmoid: bool = True,
custom_grid: typing.Optional[np.ndarray] = None
) -> np.ndarray:
return np.array([
np.array([self.get_response(phi_deg, theta_deg % 180, coef=coef, use_sigmoid=use_sigmoid), np.ones(len(self.cells)) * phi_deg,
np.ones(len(self.cells)) * theta_deg]).swapaxes(0, 1)
for phi_deg, theta_deg in (np.random.uniform(0, 360, (n, 2)) if custom_grid is None else custom_grid)
]) + np.random.normal(
0, [noise_sigma, 0, 0],
(n if custom_grid is None else len(custom_grid), len(self.cells), 3))