import sympy as sp
from brian2.parsing.sympytools import sympy_to_str, str_to_sympy
from .base import (StateUpdateMethod, UnsupportedEquationsException,
extract_method_options)
__all__ = ['exponential_euler']
[docs]def get_conditionally_linear_system(eqs, variables=None):
'''
Convert equations into a linear system using sympy.
Parameters
----------
eqs : `Equations`
The model equations.
Returns
-------
coefficients : dict of (sympy expression, sympy expression) tuples
For every variable x, a tuple (M, B) containing the coefficients M and
B (as sympy expressions) for M * x + B
Raises
------
ValueError
If one of the equations cannot be converted into a M * x + B form.
Examples
--------
>>> from brian2 import Equations
>>> eqs = Equations("""
... dv/dt = (-v + w**2) / tau : 1
... dw/dt = -w / tau : 1
... """)
>>> system = get_conditionally_linear_system(eqs)
>>> print(system['v'])
(-1/tau, w**2.0/tau)
>>> print(system['w'])
(-1/tau, 0)
'''
diff_eqs = eqs.get_substituted_expressions(variables)
coefficients = {}
for name, expr in diff_eqs:
var = sp.Symbol(name, real=True)
# Factor out the variable
s_expr = sp.collect(str_to_sympy(expr.code, variables).expand(),
var, evaluate=False)
if len(s_expr) > 2 or var not in s_expr:
raise ValueError(('The expression "%s", defining the variable %s, '
'could not be separated into linear components') %
(expr, name))
coefficients[name] = (s_expr[var], s_expr.get(1, 0))
return coefficients
[docs]class ExponentialEulerStateUpdater(StateUpdateMethod):
'''
A state updater for conditionally linear equations, i.e. equations where
each variable only depends linearly on itself (but possibly non-linearly
on other variables). Typical Hodgkin-Huxley equations fall into this
category, it is therefore the default integration method used in the
GENESIS simulator, for example.
'''
[docs] def __call__(self, equations, variables=None, method_options=None):
method_options = extract_method_options(method_options, {})
if equations.is_stochastic:
raise UnsupportedEquationsException('Cannot solve stochastic '
'equations with this state '
'updater.')
# Try whether the equations are conditionally linear
try:
system = get_conditionally_linear_system(equations, variables)
except ValueError:
raise UnsupportedEquationsException('Can only solve conditionally '
'linear systems with this '
'state updater.')
code = []
for var, (A, B) in system.iteritems():
s_var = sp.Symbol(var)
s_dt = sp.Symbol('dt')
if A == 0:
update_expression = s_var + s_dt * B
elif B != 0:
BA = B / A
# Avoid calculating B/A twice
BA_name = '_BA_' + var
s_BA = sp.Symbol(BA_name)
code += [BA_name + ' = ' + sympy_to_str(BA)]
update_expression = (s_var + s_BA)*sp.exp(A*s_dt) - s_BA
else:
update_expression = s_var*sp.exp(A*s_dt)
# The actual update step
update = '_{var} = {expr}'
code += [update.format(var=var, expr=sympy_to_str(update_expression))]
# Replace all the variables with their updated value
for var in system:
code += ['{var} = _{var}'.format(var=var)]
return '\n'.join(code)
# Copy doc from parent class
__call__.__doc__ = StateUpdateMethod.__call__.__doc__
exponential_euler = ExponentialEulerStateUpdater()