# Source code for brian2.stateupdaters.exact

```"""
Exact integration for linear equations.
"""

import itertools

import sympy as sp
from sympy import I, Symbol, Wild, im, re

from brian2.equations.codestrings import is_constant_over_dt
from brian2.parsing.sympytools import str_to_sympy, sympy_to_str
from brian2.stateupdaters.base import (
StateUpdateMethod,
UnsupportedEquationsException,
extract_method_options,
)
from brian2.utils.caching import cached
from brian2.utils.logger import get_logger

__all__ = ["linear", "exact", "independent"]

logger = get_logger(__name__)

[docs]
def get_linear_system(eqs, variables):
"""
Convert equations into a linear system using sympy.

Parameters
----------
eqs : `Equations`
The model equations.

Returns
-------
(diff_eq_names, coefficients, constants) : (list of str, `sympy.Matrix`, `sympy.Matrix`)
A tuple containing the variable names (`diff_eq_names`) corresponding
to the rows of the matrix `coefficients` and the vector `constants`,
representing the system of equations in the form M * X + B

Raises
------
ValueError
If the equations cannot be converted into an M * X + B form.
"""
diff_eqs = eqs.get_substituted_expressions(variables)
diff_eq_names = [name for name, _ in diff_eqs]

symbols = [Symbol(name, real=True) for name in diff_eq_names]

coefficients = sp.zeros(len(diff_eq_names))
constants = sp.zeros(len(diff_eq_names), 1)

for row_idx, (name, expr) in enumerate(diff_eqs):
s_expr = str_to_sympy(expr.code, variables).expand()

current_s_expr = s_expr
for col_idx, symbol in enumerate(symbols):
current_s_expr = current_s_expr.collect(symbol)
constant_wildcard = Wild("c", exclude=[symbol])
factor_wildcard = Wild(f"c_{name}", exclude=symbols)
one_pattern = factor_wildcard * symbol + constant_wildcard
matches = current_s_expr.match(one_pattern)
if matches is None:
raise UnsupportedEquationsException(
f"The expression '{expr}', "
"defining the variable "
f"'{name}', could not be "
"separated into linear "
"components."
)

coefficients[row_idx, col_idx] = matches[factor_wildcard]
current_s_expr = matches[constant_wildcard]

# The remaining constant should be a true constant
constants[row_idx] = current_s_expr

return (diff_eq_names, coefficients, constants)

[docs]
class IndependentStateUpdater(StateUpdateMethod):
"""
A state update for equations that do not depend on other state variables,
i.e. 1-dimensional differential equations. The individual equations are
solved by sympy.

.. deprecated:: 2.1
This method might be removed from future versions of Brian.
"""

[docs]
def __call__(self, equations, variables=None, method_options=None):
logger.warn(
"The 'independent' state updater is deprecated and might be "
"removed in future versions of Brian.",
"deprecated_independent",
once=True,
)
extract_method_options(method_options, {})
if equations.is_stochastic:
raise UnsupportedEquationsException(
"Cannot solve stochastic equations with this state updater"
)
if variables is None:
variables = {}

diff_eqs = equations.get_substituted_expressions(variables)

t = Symbol("t", real=True, positive=True)
dt = Symbol("dt", real=True, positive=True)
t0 = Symbol("t0", real=True, positive=True)

code = []
for name, expression in diff_eqs:
rhs = str_to_sympy(expression.code, variables)

# We have to be careful and use the real=True assumption as well,
# otherwise sympy doesn't consider the symbol a match to the content
# of the equation
var = Symbol(name, real=True)
f = sp.Function(name)
rhs = rhs.subs(var, f(t))
derivative = sp.Derivative(f(t), t)
diff_eq = sp.Eq(derivative, rhs)
# TODO: simplify=True sometimes fails with 0.7.4, see:
# https://github.com/sympy/sympy/issues/2666
try:
general_solution = sp.dsolve(diff_eq, f(t), simplify=True)
except RuntimeError:
general_solution = sp.dsolve(diff_eq, f(t), simplify=False)
# Check whether this is an explicit solution
if not getattr(general_solution, "lhs", None) == f(t):
raise UnsupportedEquationsException(
f"Cannot explicitly solve: {str(diff_eq)}"
)
# Solve for C1 (assuming "var" as the initial value and "t0" as time)
if general_solution.has(Symbol("C1")):
if general_solution.has(Symbol("C2")):
raise UnsupportedEquationsException(
f"Too many constants in solution: {str(general_solution)}"
)
constant_solution = sp.solve(general_solution, Symbol("C1"))
if len(constant_solution) != 1:
raise UnsupportedEquationsException(
"Couldn't solve for the constant C1 in : %s "
% str(general_solution)
)
constant = constant_solution[0].subs(t, t0).subs(f(t0), var)
solution = general_solution.rhs.subs("C1", constant)
else:
solution = general_solution.rhs.subs(t, t0).subs(f(t0), var)
# Evaluate the expression for one timestep
solution = solution.subs(t, t + dt).subs(t0, t)
# only try symplifying it -- it sometimes raises an error
try:
solution = solution.simplify()
except ValueError:
pass

code.append(f"{name} = {sympy_to_str(solution)}")

return "\n".join(code)

[docs]
class LinearStateUpdater(StateUpdateMethod):
"""
A state updater for linear equations. Derives a state updater step from the
analytical solution given by sympy. Uses the matrix exponential (which is
only implemented for diagonalizable matrices in sympy).
"""

[docs]
@cached
def __call__(self, equations, variables=None, method_options=None):
method_options = extract_method_options(method_options, {"simplify": True})

if equations.is_stochastic:
raise UnsupportedEquationsException(
"Cannot solve stochastic equations with this state updater."
)
if variables is None:
variables = {}

# Get a representation of the ODE system in the form of
# dX/dt = M*X + B
varnames, matrix, constants = get_linear_system(equations, variables)

# No differential equations, nothing to do (this occurs sometimes in the
# test suite where the whole model is nothing more than something like
# 'v : 1')
if matrix.shape == (0, 0):
return ""

# Make sure that the matrix M is constant, i.e. it only contains
# external variables or constant variables

# Check for time dependence
dt_value = variables["dt"].get_value()[0] if "dt" in variables else None

# This will raise an error if we meet the symbol "t" anywhere
# except as an argument of a locally constant function
for entry in itertools.chain(matrix, constants):
if not is_constant_over_dt(entry, variables, dt_value):
raise UnsupportedEquationsException(
f"Expression '{sympy_to_str(entry)}' is not guaranteed to be "
"constant over a time step."
)

symbols = [Symbol(variable, real=True) for variable in varnames]
solution = sp.solve_linear_system(matrix.row_join(constants), *symbols)
if solution is None or set(symbols) != set(solution.keys()):
raise UnsupportedEquationsException(
"Cannot solve the given equations with this stateupdater."
)
b = sp.ImmutableMatrix([solution[symbol] for symbol in symbols])

# Solve the system
dt = Symbol("dt", real=True, positive=True)
try:
A = (matrix * dt).exp()
except NotImplementedError:
raise UnsupportedEquationsException(
"Cannot solve the given equations with this stateupdater."
)
if method_options["simplify"]:
A = A.applyfunc(lambda x: sp.factor_terms(sp.cancel(sp.signsimp(x))))
C = sp.ImmutableMatrix(A * b) - b
_S = sp.MatrixSymbol("_S", len(varnames), 1)
updates = A * _S + C

# The solution contains _S[0, 0], _S[1, 0] etc. for the state variables,
# replace them with the state variable names
abstract_code = []
for variable, update in zip(varnames, updates):
rhs = update
if rhs.has(I, re, im):
raise UnsupportedEquationsException(
"The solution to the linear system "
"contains complex values "
"which is currently not implemented."
)
for row_idx, varname in enumerate(varnames):
rhs = rhs.subs(_S[row_idx, 0], varname)

# Do not overwrite the real state variables yet, the update step
# of other state variables might still need the original values
abstract_code.append(f"_{variable} = {sympy_to_str(rhs)}")

# Update the state variables
for variable in varnames:
abstract_code.append(f"{variable} = _{variable}")
return "\n".join(abstract_code)

def __repr__(self):
return f"{self.__class__.__name__}()"

independent = IndependentStateUpdater()
linear = LinearStateUpdater()
exact = linear
```