# Source code for brian2.stateupdaters.exponential_euler

```
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**2.0) / 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)

s_expr = str_to_sympy(expr.code, variables).expand()
if s_expr.has(var):
# Factor out the variable
s_expr = sp.collect(s_expr,
var, evaluate=False)

if len(s_expr) > 2 or var not in s_expr:
raise ValueError(f"The expression '{expr}', defining the variable "
f"'{name}', could not be separated into linear "
"components.")
coefficients[name] = (s_expr[var], s_expr.get(1, 0))
else:
coefficients[name] = (0, s_expr)

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):
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.items():
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 = f"_BA_{var}"
s_BA = sp.Symbol(BA_name)
code += [f"{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
code += [f'_{var} = {sympy_to_str(update_expression)}']

# Replace all the variables with their updated value
for var in system:
code += [f'{var} = _{var}']

return '\n'.join(code)
# Copy doc from parent class
__call__.__doc__ = StateUpdateMethod.__call__.__doc__

exponential_euler = ExponentialEulerStateUpdater()

```