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 "
f"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
updates = updates.as_explicit()
# The solution contains _S[0, 0], _S[1, 0] etc. for the state variables,
# replace them with the state variable names
abstract_code = []
for idx, (variable, update) in enumerate(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 __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
t = Symbol('t', real=True, positive=True)
# 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(
('Expression "{}" is not guaranteed to be constant over a '
'time step').format(sympy_to_str(entry)))
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
updates = updates.as_explicit()
# The solution contains _S[0, 0], _S[1, 0] etc. for the state variables,
# replace them with the state variable names
abstract_code = []
for idx, (variable, update) in enumerate(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('_' + variable + ' = ' + sympy_to_str(rhs))
# Update the state variables
for variable in varnames:
abstract_code.append('{variable} = _{variable}'.format(variable=variable))
return '\n'.join(abstract_code)
def __call__(self, eqs, variables=None, method_options=None):
'''
Apply a state updater description to model equations.
Parameters
----------
eqs : `Equations`
The equations describing the model
variables: dict-like, optional
The `Variable` objects for the model. Ignored by the explicit
state updater.
method_options : dict, optional
Additional options to the state updater (not used at the moment
for the explicit state updaters).
Examples
--------
>>> from brian2 import *
>>> eqs = Equations('dv/dt = -v / tau : volt')
>>> print(euler(eqs))
_v = -dt*v/tau + v
v = _v
>>> print(rk4(eqs))
__k_1_v = -dt*v/tau
__k_2_v = -dt*(__k_1_v/2 + v)/tau
__k_3_v = -dt*(__k_2_v/2 + v)/tau
__k_4_v = -dt*(__k_3_v + v)/tau
_v = __k_1_v/6 + __k_2_v/3 + __k_3_v/3 + __k_4_v/6 + v
v = _v
'''
extract_method_options(method_options, {})
# Non-stochastic numerical integrators should work for all equations,
# except for stochastic equations
if eqs.is_stochastic and self.stochastic is None:
raise UnsupportedEquationsException('Cannot integrate '
'stochastic equations with '
'this state updater.')
if self.custom_check:
self.custom_check(eqs, variables)
# The final list of statements
statements = []
stochastic_variables = eqs.stochastic_variables
# The variables for the intermediate results in the state updater
# description, e.g. the variable k in rk2
intermediate_vars = [var for var, expr in self.statements]
# A dictionary mapping all the variables in the equations to their
# sympy representations
eq_variables = dict(((var, _symbol(var)) for var in eqs.eq_names))
# Generate the random numbers for the stochastic variables
for stochastic_variable in stochastic_variables:
statements.append(stochastic_variable + ' = ' + 'dt**.5 * randn()')
substituted_expressions = eqs.get_substituted_expressions(variables)
# Process the intermediate statements in the stateupdater description
for intermediate_var, intermediate_expr in self.statements:
# Split the expression into a non-stochastic and a stochastic part
non_stochastic_expr, stochastic_expr = split_expression(intermediate_expr)
# Execute the statement by appropriately replacing the functions f
# and g and the variable x for every equation in the model.
# We use the model equations where the subexpressions have
# already been substituted into the model equations.
for var, expr in substituted_expressions:
for xi in stochastic_variables:
RHS = self._generate_RHS(eqs, var, eq_variables, intermediate_vars,
expr, non_stochastic_expr,
stochastic_expr, xi)
statements.append(intermediate_var+'_'+var+'_'+xi+' = '+RHS)
if not stochastic_variables: # no stochastic variables
RHS = self._generate_RHS(eqs, var, eq_variables, intermediate_vars,
expr, non_stochastic_expr,
stochastic_expr)
statements.append(intermediate_var+'_'+var+' = '+RHS)
# Process the "return" line of the stateupdater description
non_stochastic_expr, stochastic_expr = split_expression(self.output)
if eqs.is_stochastic and (self.stochastic != 'multiplicative' and
eqs.stochastic_type == 'multiplicative'):
# The equations are marked as having multiplicative noise and the
# current state updater does not support such equations. However,
# it is possible that the equations do not use multiplicative noise
# at all. They could depend on time via a function that is constant
# over a single time step (most likely, a TimedArray). In that case
# we can integrate the equations
dt_value = variables['dt'].get_value()[0] if 'dt' in variables else None
for _, expr in substituted_expressions:
_, stoch = expr.split_stochastic()
if stoch is None:
continue
# There could be more than one stochastic variable (e.g. xi_1, xi_2)
for _, stoch_expr in stoch.items():
sympy_expr = str_to_sympy(stoch_expr.code)
# The equation really has multiplicative noise, if it depends
# on time (and not only via a function that is constant
# over dt), or if it depends on another variable defined
# via differential equations.
if (not is_constant_over_dt(sympy_expr, variables, dt_value)
or len(stoch_expr.identifiers & eqs.diff_eq_names)):
raise UnsupportedEquationsException('Cannot integrate '
'equations with '
'multiplicative noise with '
'this state updater.')
# Assign a value to all the model variables described by differential
# equations
for var, expr in substituted_expressions:
RHS = self._generate_RHS(eqs, var, eq_variables, intermediate_vars,
expr, non_stochastic_expr, stochastic_expr,
stochastic_variables)
statements.append('_' + var + ' = ' + RHS)
# Assign everything to the final variables
for var, expr in substituted_expressions:
statements.append(var + ' = ' + '_' + var)
return '\n'.join(statements)