/usr/share/pyshared/brian/equations.py is in python-brian 1.4.1-2.
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# Copyright ENS, INRIA, CNRS
# Contributors: Romain Brette (brette@di.ens.fr) and Dan Goodman (goodman@di.ens.fr)
#
# Brian is a computer program whose purpose is to simulate models
# of biological neural networks.
#
# This software is governed by the CeCILL license under French law and
# abiding by the rules of distribution of free software. You can use,
# modify and/ or redistribute the software under the terms of the CeCILL
# license as circulated by CEA, CNRS and INRIA at the following URL
# "http://www.cecill.info".
#
# As a counterpart to the access to the source code and rights to copy,
# modify and redistribute granted by the license, users are provided only
# with a limited warranty and the software's author, the holder of the
# economic rights, and the successive licensors have only limited
# liability.
#
# In this respect, the user's attention is drawn to the risks associated
# with loading, using, modifying and/or developing or reproducing the
# software by the user in light of its specific status of free software,
# that may mean that it is complicated to manipulate, and that also
# therefore means that it is reserved for developers and experienced
# professionals having in-depth computer knowledge. Users are therefore
# encouraged to load and test the software's suitability as regards their
# requirements in conditions enabling the security of their systems and/or
# data to be ensured and, more generally, to use and operate it in the
# same conditions as regards security.
#
# The fact that you are presently reading this means that you have had
# knowledge of the CeCILL license and that you accept its terms.
# ----------------------------------------------------------------------------------
#
'''
Differential equations for Brian models.
'''
#from scipy.weave import blitz
from operator import isSequenceType
import types
from units import *
from stdunits import *
from inspection import *
from scipy import exp
from scipy import weave
from globalprefs import *
import re
import inspect
import optimiser
import warnings
import uuid
from numpy import zeros, ones
import numpy
from log import *
from optimiser import *
from scipy import optimize
import unitsafefunctions
import copy
try:
import sympy
use_sympy = True
except:
warnings.warn('sympy not installed')
use_sympy = False
__all__ = ['Equations', 'unique_id']
# TODO: write interface for equations.py
def unique_id():
"""
Returns a unique name (e.g. for internal hidden variables).
"""
return '_' + str(uuid.uuid1().int)
class Equations(object):
"""Container that stores equations from which models can be created
Initialised as::
Equations(expr[,level=0[,keywords...]])
with arguments:
``expr``
An expression, which can each be a string representing equations,
an :class:`Equations` objects, or a list of strings and :class:`Equations` objects.
See below for details of the string format.
``level``
Indicates how many levels back in the stack the namespace for string
equations is found, so that e.g. ``level=0`` looks in the
namespace of the function where the :class:`Equations` object was created,
``level=1`` would look in the namespace of the function that called the
function where the :class:`Equations` object was created, etc.
Normally you can just leave this out.
``keywords``
Any sequence of keyword pairs ``key=value`` where the string ``key``
in the string equations will be replaced with ``value`` which can
be either a string, value or ``None``, in the latter case a unique
name will be generated automatically (but it won't be pretty).
Systems of equations can be defined by passing lists of :class:`Equations` to a
new :class:`Equations` object, or by adding :class:`Equations` objects together (the usage
is similar to that of a Python ``list``).
**String equations**
String equations can be of any of the following forms:
(1) ``dx/dt = f : unit`` (differential equation)
(2) ``x = f : unit`` (equation)
(3) ``x = y`` (alias)
(4) ``x : unit`` (parameter)
Here each of ``x`` and ``y`` can be any valid Python variable name,
``f`` can be any valid Python expression, and ``unit`` should be the
unit of the corresponding ``x``. You can also include multi-line
expressions by appending a ``\`` character at the end of each line
which is continued on the next line (following the Python standard),
or comments by including a ``#`` symbol.
These forms mean:
*Differential equation*
A differential equation with variable ``x`` which has physical
units ``unit``. The variable ``x`` will become one of the state
variables of the model.
*Equation*
An equation defining the meaning of ``x`` can be used for building
systems of complicated differential equations.
*Alias*
The variable ``x`` becomes equivalent to the variable ``y``, useful
for connecting two separate systems of equations together.
*Parameter*
The variable ``x`` will have physical units ``unit`` and will be
one of the state variables of the model (but will not evolve
dynamically, instead it should be set by the user).
.. index::
single: xi
pair: xi; noise
single: white noise
single: gaussian noise
single: noise
single: noise; gaussian
single: noise; white
**Noise**
String equations can also use the reserved term ``xi`` for a
Gaussian white noise with mean 0 and variance 1.
**Example usage** ::
eqs=Equations('''
dv/dt=(u-v)/tau : volt
u=3*v : volt
w=v
''')
**Details**
For more details, see :ref:`moreonequations` in the user manual.
"""
def __init__(self, expr='', level=0, **kwds):
# Empty object
self._Vm = None # name of variable with membrane potential
self._eq_names = [] # equations names
self._diffeq_names = [] # differential equations names
self._diffeq_names_nonzero = [] # differential equations names
self._function = {} # dictionary of functions
self._string = {} # dictionary of strings (defining the functions)
self._namespace = {} # dictionary of namespaces for the strings (globals,locals)
self._alias = {} # aliases (mapping name1 -> name2)
self._units = {'t':second} # dictionary of units
self._dependencies = {} # dictionary of dependencies (on static equations)
self._useweave = get_global_preference('useweave')
self._cpp_compiler = get_global_preference('weavecompiler')
self._extra_compile_args = ['-O3']
if self._cpp_compiler == 'gcc':
self._extra_compile_args += get_global_preference('gcc_options') # ['-march=native', '-ffast-math']
self._frozen = False # True if all units and parameters are gone
self._prepared = False
if expr is None:
expr = ''
if not isinstance(expr, str): # assume it is a sequence of Equations objects
for eqs in expr:
if not isinstance(eqs, Equations):
eqs = Equations(eqs, level=level + 1)
self += eqs
elif expr != '':
# Check keyword arguments
param_dict = {}
for name, value in kwds.iteritems():
if value is None: # name is not important: choose unique name
value = unique_id()
if isinstance(value, str): # variable name substitution
expr = re.sub('\\b' + name + '\\b', value, expr)
expr = re.sub('\\bd' + name + '\\b', 'd' + value, expr) # derivative
else:
param_dict[name] = value
if kwds == {}: # weird: changed from param_dict on 18/06/08
self.parse_string_equations(expr, level=level + 1)
else:
self.parse_string_equations(expr, namespace=param_dict, level=level + 1)
"""
-----------------------------------------------------------------------
PARSING AND BUILDING NAMESPACES
-----------------------------------------------------------------------
"""
def parse_string_equations(self, eqns, level=1, namespace=None):
"""
Parses a string defining equations and builds an Equations object.
Uses the namespace in the given level of the stack.
"""
diffeq_pattern = re.compile('\s*d(\w+)\s*/\s*dt\s*=\s*(.+?)\s*:\s*(.*)')
eq_pattern = re.compile('\s*(\w+)\s*=\s*(.+?)\s*:\s*(.*)')
alias_pattern = re.compile('\s*(\w+)\s*=\s*(\w+)\s*$')
param_pattern = re.compile('\s*(\w+)\s*:\s*(.*)')
empty_pattern = re.compile('\s*$')
patterns = [diffeq_pattern, eq_pattern, alias_pattern, param_pattern, empty_pattern]
# Merge multi-line statements
eqns = re.sub('\\\s*?\n', ' ', eqns)
# Namespace of the functions
ns_global, ns_local = namespace, namespace
if namespace is None:
frame = inspect.stack()[level + 1][0]
ns_global, ns_local = frame.f_globals, frame.f_locals
#print frame.f_code.co_filename #useful for debugging which file the namespace came from
for line in eqns.splitlines():
line = re.sub('#.*', '', line) # remove comments
result = None
for pattern in patterns:
result = pattern.match(line)
if result:
break
if result == None:
raise TypeError, "Invalid equation string: " + line
if pattern == eq_pattern:
name, eq, unit = result.groups()
self.add_eq(name, eq, unit, ns_global, ns_local)
elif pattern == diffeq_pattern:
name, eq, unit = result.groups()
self.add_diffeq(name, eq, unit, ns_global, ns_local)
elif pattern == alias_pattern:
name1, name2 = result.groups()
self.add_alias(name1, name2)
elif pattern == param_pattern:
name, unit = result.groups()
self.add_param(name, unit, ns_global, ns_local)
def add_eq(self, name, eq, unit, global_namespace={}, local_namespace={}):
"""
Inserts an equation.
name = variable name
eq = string definition
unit = unit of the variable (possibly a string)
*_namespace = namespaces associated to the string
"""
# Find external objects
vars = list(get_identifiers(eq))
if type(unit) == types.StringType:
vars.extend(list(get_identifiers(unit)))
self._namespace[name] = {}
for var in vars:
if var in local_namespace: #local
self._namespace[name][var] = local_namespace[var]
elif var in global_namespace: #global
self._namespace[name][var] = global_namespace[var]
elif var in globals(): # typically units
self._namespace[name][var] = globals()[var]
self._eq_names.append(name)
if type(unit) == types.StringType:
self._units[name] = eval(unit, self._namespace[name].copy())
else:
self._units[name] = unit
self._string[name] = eq
def add_diffeq(self, name, eq, unit, global_namespace={}, local_namespace={}, nonzero=True):
"""
Inserts a differential equation.
name = variable name
eq = string definition
unit = unit of the variable (possibly a string)
*_namespace = namespaces associated to the string
nonzero = False if dx/dt=0 (parameter)
"""
# Find external objects
vars = list(get_identifiers(eq))
if type(unit) == types.StringType:
vars.extend(list(get_identifiers(unit)))
self._namespace[name] = {}
for var in vars:
if var in local_namespace: #local
self._namespace[name][var] = local_namespace[var]
elif var in global_namespace: #global
self._namespace[name][var] = global_namespace[var]
elif var in globals(): # typically units
self._namespace[name][var] = globals()[var]
self._diffeq_names.append(name)
if type(unit) == types.StringType:
self._units[name] = eval(unit, self._namespace[name].copy())
else:
self._units[name] = unit
self._string[name] = eq
if nonzero:
self._diffeq_names_nonzero.append(name)
def add_alias(self, name1, name2):
"""
Inserts an alias.
name1 = new name
name2 = old name
"""
self._alias[name1] = name2
# TODO: what if name2 is not defined yet?
self.add_eq(name1, name2, self._units[name2])
def add_param(self, name, unit, global_namespace={}, local_namespace={}):
"""
Inserts a parameter.
name = variable name
eq = string definition
unit = unit of the variable (possibly a string)
*_namespace = namespaces associated to the string
"""
if isinstance(unit, Quantity):
unit = scalar_representation(unit)
self.add_diffeq(name, '0*' + unit + '/second', unit, global_namespace, local_namespace, nonzero=False)
"""
-----------------------------------------------------------------------
FINALISATION
-----------------------------------------------------------------------
"""
def prepare(self, check_units=True):
'''
Do a number of checks (units) and preparation of the object.
'''
if self._prepared:
return
# Let Vm be the first differential equation
vm_name = self.get_Vm()
if vm_name:
# TODO: INFO logging
i = self._diffeq_names.index(vm_name)
self._diffeq_names[0], self._diffeq_names[i] = self._diffeq_names[i], self._diffeq_names[0]
else:
pass
# TODO: WARNING log that a potential problem has occurred here?
# Clean namespace (avoids conflicts between variables and external variables)
self.clean_namespace()
# Compile strings to functions
self.compile_functions()
# Check units
if check_units: self.check_units()
# Set the update order of (static) variables
self.set_eq_order()
# Replace static variables by their value in differential equations
self.substitute_eq()
self.compile_functions()
# Check free variables
free_vars = self.free_variables()
if free_vars != []:
log_info('brian.equations', 'Free variables: ' + str(free_vars))
self._prepared = True
def get_Vm(self):
'''
Finds the variable that is most likely to be the
membrane potential.
'''
if self._Vm:
return self._Vm
vm_names = ['v', 'V', 'vm', 'Vm']
guesses = [var for var in self._diffeq_names if var in vm_names]
if len(guesses) == 1: # Unambiguous
return guesses[0]
else: # Ambiguous or not found
return None
def clean_namespace(self):
'''
Removes all variable names from namespaces
'''
all_variables = self._eq_names + self._diffeq_names + self._alias.keys() + ['t']
for name in self._namespace:
for var in all_variables:
if var in self._namespace[name]:
log_warn('brian.equations', 'Equation variable ' + var + ' also exists in the namespace')
del self._namespace[name][var]
def compile_functions(self, freeze=False):
"""
Compile all functions defined as strings.
If freeze is True, all external parameters and units are replaced by their value.
ALL FUNCTIONS MUST HAVE STRINGS.
"""
all_variables = self._eq_names + self._diffeq_names + self._alias.keys() + ['t']
# Check if freezable
freeze = freeze and all([optimiser.freeze(expr, all_variables, self._namespace[name])\
for name, expr in self._string.iteritems()])
self._frozen = freeze
# Compile strings to functions
for name, expr in self._string.iteritems():
namespace = self._namespace[name] # name space of the function
# Find variables
vars = [var for var in get_identifiers(expr) if var in all_variables]
if freeze:
expr = optimiser.freeze(expr, all_variables, namespace)
#self._string[name]=expr # should we?
#namespace={}
s = "lambda " + ','.join(vars) + ":" + expr
self._function[name] = eval(s, namespace)
def check_units(self):
'''
Checks the units of the differential equations, using
the units of x.
dx_i/dt must have units of x_i / time.
'''
self.set_eq_order()
# Better: replace xi in the string, or in the namespace
try:
for var in self._eq_names:
f = self._function[var]
old_func_globals = copy.copy(f.func_globals)
f.func_globals['xi'] = 0 * second ** -.5 # Noise
f.func_globals.update(namespace_replace_quantity_with_pure(f.func_globals))
units = namespace_replace_quantity_with_pure(self._units)
self.apply(var, units) + self._units[var] # Check that the two terms have the same dimension
f.func_globals.update(old_func_globals)
for var in self._diffeq_names:
f = self._function[var]
old_func_globals = copy.copy(f.func_globals)
f.func_globals['xi'] = 0 * second ** -.5 # Noise
f.func_globals.update(namespace_replace_quantity_with_pure(f.func_globals))
units = namespace_replace_quantity_with_pure(self._units)
self.apply(var, units) + (self._units[var] / second) # Check that the two terms have the same dimension
f.func_globals.update(old_func_globals)
except DimensionMismatchError, inst:
raise DimensionMismatchError("The differential equation of " + var + " is not homogeneous", *inst._dims)
except:
warnings.warn("Unexpected exception in checking units of " + var)
raise
def set_eq_order(self):
'''
Computes the internal depency graph of static variables
and deduces the update order.
Sets the list of dependencies of dynamic variables on static variables.
This is called by check_units()
'''
if len(self._eq_names) > 0:
# Internal dependency dictionary
dependency = {}
for key in self._eq_names:
f = self._function[key]
dependency[key] = [var for var in f.func_code.co_varnames if var in self._eq_names]
# Sets the order
staticvars_list = []
no_dep = None
while (len(staticvars_list) < len(self._eq_names)) and (no_dep != []):
no_dep = [key for key, value in dependency.iteritems() if value == []]
staticvars_list += no_dep
# Clear dependency list
for key in no_dep:
del dependency[key]
for key, value in dependency.iteritems():
dependency[key] = [var for var in value if not(var in staticvars_list)]
if no_dep == []: # The dependency graph has cycles!
raise ReferenceError, "The static variables are referring to each other"
else:
staticvars_list = []
# Calculate dependencies on static variables
self._dependencies = {}
for key in staticvars_list:
self._dependencies[key] = []
for var in self._function[key].func_code.co_varnames:
if var in self._eq_names:
self._dependencies[key] += [var] + self._dependencies[var]
for key in self._diffeq_names:
f = self._function[key]
self._dependencies[key] = []
for var in f.func_code.co_varnames:
if var in self._eq_names:
self._dependencies[key] += [var] + self._dependencies[var]
# Sort the dependency lists
for key in self._dependencies:
staticdep = [(staticvars_list.index(var), var) for var in self._dependencies[key]]
staticdep.sort()
self._dependencies[key] = [x[1] for x in staticdep]
# Update _eq
self._eq_names = staticvars_list
def substitute_eq(self, name=None):
"""
Replaces the static variable 'name' by its value in differential
equations.
If None: substitute all static variables.
"""
if name is None:
for var in self._eq_names[-1::-1]: # reverse order
self.substitute_eq(var)
else:
self.add_prefix_namespace(name)
#print name
for var in self._diffeq_names_nonzero:
# String
self._string[var] = re.sub("\\b" + name + "\\b", '(' + self._string[name] + ')', self._string[var])
# Namespace
self._namespace[var].update(self._namespace[name])
#print self
def add_prefix_namespace(self, name):
"""
Make the variables in the namespace associated to variable name
specific to that variable by inserting the prefix name_.
"""
vars = self._namespace[name].keys()
untransformed_funcs = set(getattr(unitsafefunctions, v) for v in unitsafefunctions.quantity_versions)
untransformed_funcs.update(set([numpy.clip]))
for var in vars:
v = self._namespace[name][var]
addprefix = True
if isinstance(v, numpy.ufunc):
addprefix = False
try:
if v in untransformed_funcs:
addprefix = False
except TypeError: #unhashable types
pass
if addprefix:
# String
self._string[name] = re.sub("\\b" + var + "\\b", name + '_' + var, self._string[name])
# Namespace
self._namespace[name][name + '_' + var] = self._namespace[name][var]
del self._namespace[name][var]
def free_variables(self):
"""
Returns the list of free variables (i.e., which are not defined within the
equation string).
"""
all_variables = self._eq_names + self._diffeq_names + self._alias.keys() + ['t']
free_vars = []
for expr in self._string.itervalues():
free_vars += [name for name in get_identifiers(expr) if name not in all_variables]
return list(set(free_vars))
"""
-----------------------------------------------------------------------
CALCULATING RHS OF DIFF EQUATIONS
-----------------------------------------------------------------------
"""
def apply(self, state, vardict):
'''
Calculates self._function[state] with arguments in vardict and
static variables. The dictionary is filled with the required
static variables.
'''
f = self._function[state]
# Calculate static variables
for var in self._dependencies[state]:
# could add something like: if var not in vardict: this would allow you to override the dependencies if you wanted to - worth doing?
vardict[var] = call_with_dict(self._function[var], vardict)
return f(*[vardict[var] for var in f.func_code.co_varnames])
"""
-----------------------------------------------------------------------
EQUATION INSPECTION
-----------------------------------------------------------------------
"""
def is_stochastic(self, var=None):
'''
Returns True if the equation for var is stochastic,
or if all equations are stochastic (var=None).
'''
if var:
return 'xi' in get_identifiers(self._string[var])
else:
return any([self.is_stochastic(name) for name in self._diffeq_names_nonzero])
def is_time_dependent(self, var=None):
'''
Returns True if the equation for var is time dependent,
or if all equations are time dependent (var=None).
'''
if var:
return 't' in get_identifiers(self._string[var])
else:
return any([self.is_time_dependent(name) for name in self._diffeq_names_nonzero])
# return any([self.is_time_dependent(name) for name in self._diffeq_names_nonzero+self._eq_names])
def is_linear(self):
'''
Returns True if all equations are linear.
If the equations are time dependent, then returns False.
If the equations depend on external functions, then returns False.
'''
if self.is_time_dependent():
return False
for f in self._namespace.iterkeys():
if any([type(key) == types.FunctionType for key in self._namespace[f].itervalues()]):
return False
return all([is_affine(f) for f in self._function.itervalues()])
def is_conditionally_linear(self):
'''
Returns True if the differential equations are linear with respect to the
state variable.
'''
# Equations have to be prepared for it to work.
for var in self._diffeq_names:
S = self._units.copy()
S[var] = AffineFunction()
try:
self.apply(var, S)
except:
return False
return True
"""
-----------------------------------------------------------------------
NUMERICAL INTEGRATION (to be replaced by code generation)
-----------------------------------------------------------------------
"""
def forward_euler(self, S, dt):
'''
Updates the value of the state variables in dictionary S
with the forward Euler algorithm over step dt.
'''
# Calculate all static variables (or do that after?)
#for var in self._eq_names:
# S[var]=call_with_dict(self._function[var],S)
# Calculate derivatives
buffer = {}
for varname in self._diffeq_names_nonzero:
f = self._function[varname]
buffer[varname] = f(*[S[var] for var in f.func_code.co_varnames])
# Update variables
for var in self._diffeq_names_nonzero:
S[var] += dt * buffer[var]
def forward_euler_code_string(self):
'''
Generates Python code for a forward Euler step.
'''
# TODO: check if it can really be frozen
# TODO: change /a to *(1/a) with precalculation (use parser)
all_variables = self._eq_names + self._diffeq_names + self._alias.keys() + ['t']
# nonzero? insert dt?
vars_tmp = [name + '__tmp' for name in self._diffeq_names]
lines = ','.join(self._diffeq_names) + '=P._S\n'
lines += ','.join(vars_tmp) + '=P._dS\n'
for name in self._diffeq_names_nonzero:
namespace = self._namespace[name]
expr = optimiser.freeze(self._string[name], all_variables, namespace)
lines += name + '__tmp[:]=' + expr + '\n'
lines += 'P._S+=dt*P._dS\n'
#print lines
return lines
# Return a function f(P) or a namespace (exec code in namespace)
# 1st option: include directly in neurongroup._state_updater (good?)
def forward_euler_code(self):
'''
Generates Python code for a forward Euler step.
'''
# TODO: check if it can really be frozen
# TODO: change /a to *(1/a) with precalculation (use parser)
all_variables = self._eq_names + self._diffeq_names + self._alias.keys() + ['t']
# nonzero? insert dt?
vars_tmp = [name + '__tmp' for name in self._diffeq_names]
lines = ','.join(self._diffeq_names) + '=P._S\n'
lines += ','.join(vars_tmp) + '=P._dS\n'
for name in self._diffeq_names_nonzero:
namespace = self._namespace[name]
expr = optimiser.freeze(self._string[name], all_variables, namespace)
lines += name + '__tmp[:]=' + expr + '\n'
lines += 'P._S+=dt*P._dS\n'
#print lines
return compile(lines, 'Euler update code', 'exec')
# Return a function f(P) or a namespace (exec code in namespace)
# 1st option: include directly in neurongroup._state_updater (good?)
def Runge_Kutta2(self, S, dt):
'''
Updates the value of the state variables in dictionary S
with the 2nd order Runge-Kutta algorithm over step dt (midpoint).
'''
# Calculate all static variables (or do that after?)
#for var in self._eq_names:
# S[var]=call_with_dict(self._function[var],S)
# Calculate derivatives
buffer = {}
S_half = S.copy()
# Half a step
for varname in self._diffeq_names_nonzero:
f = self._function[varname]
buffer[varname] = f(*[S[var] for var in f.func_code.co_varnames])
# Update variables
for var in self._diffeq_names_nonzero:
S_half[var] = S[var] + .5 * dt * buffer[var]
# Whole step
for varname in self._diffeq_names_nonzero:
f = self._function[varname]
buffer[varname] = f(*[S_half[var] for var in f.func_code.co_varnames])
# Update variables
for var in self._diffeq_names_nonzero:
S[var] += dt * buffer[var]
def exponential_euler(self, S, dt):
'''
Updates the value of the state variables in dictionary S
with an exponential Euler algorithm over step dt.
Test with is_conditionally_linear first.
Same as default integration method in Genesis.
Close to the implicit Euler method in Neuron.
'''
# Calculate all static variables (BAD: INSERT IT BELOW)
#for var in self._eq_names:
# S[var]=call_with_dict(self._function[var],S)
n = len(S[self._diffeq_names_nonzero[0]])
# Calculate the coefficients of the affine function
Z = zeros(n)
O = ones(n)
A = {}
B = {}
for varname in self._diffeq_names_nonzero:
f = self._function[varname]
oldval = S[varname]
S[varname] = Z
B[varname] = f(*[S[var] for var in f.func_code.co_varnames]).copy() # important if compiled
S[varname] = O
A[varname] = f(*[S[var] for var in f.func_code.co_varnames]) - B[varname]
B[varname] /= A[varname]
S[varname] = oldval
# Integrate
for varname in self._diffeq_names_nonzero:
f = self._function[varname]
if self._useweave:
Bx = B[varname]
Ax = A[varname]
Sx = S[varname]
# Compilation with blitz: we need an approximation because exp is not understood
#weave.blitz('Sx[:]=-Bx+(Sx+Bx)*(1.+Ax*dt*(1.+.5*Ax*dt))',check_size=0)
code = """
for(int k=0;k<n;k++)
Sx(k)=-Bx(k)+(Sx(k)+Bx(k))*exp(Ax(k)*dt);
"""
weave.inline(code, ['n', 'Bx', 'Sx', 'Ax', 'dt'], \
compiler=self._cpp_compiler,
type_converters=weave.converters.blitz,
extra_compile_args=self._extra_compile_args)
else:
#S[varname][:]=-B[varname]+(S[varname]+B[varname])*exp(A[varname]*dt)
# A little faster:
S[varname] += B[varname]
S[varname] *= exp(A[varname] * dt)
S[varname] -= B[varname]
def exponential_euler_code(self):
'''
Generates Python code for an exponential Euler step.
Not efficient for the moment!
'''
all_variables = self._eq_names + self._diffeq_names + self._alias.keys() + ['t']
vars_tmp = [name + '__tmp' for name in self._diffeq_names]
lines = ','.join(self._diffeq_names) + '=P._S\n'
lines += ','.join(vars_tmp) + '=P._dS\n'
for name in self._diffeq_names:
# Freeze
namespace = self._namespace[name]
expr = optimiser.freeze(self._string[name], all_variables, namespace)
# Find a and b in dx/dt=a*x+b
sym_expr = symbolic_eval(expr)
if isinstance(sym_expr, float):
lines += name + '__tmp[:]=' + name + '+(' + str(expr) + ')*dt\n'
else:
sym_expr = sym_expr.expand()
sname = sympy.Symbol(name)
terms = sympy.collect(sym_expr, name, evaluate=False)
if sname ** 0 in terms:
b = terms[sname ** 0]
else:
b = 0
if sname in terms:
a = terms[sname]
else:
a = 0
lines += name + '__tmp[:]=' + str(-b / a + (sname + b / a) * sympy.exp(a * sympy.Symbol('dt'))) + '\n'
lines += 'P._S[:]=P._dS'
#print lines
return compile(lines, 'Exponential Euler update code', 'exec')
"""
-------------------
COMBINING EQUATIONS
-------------------
"""
def __add__(self, other):
'''
Union of two sets of equations
'''
if not isinstance(other, Equations):
other = Equations(other, level=1)
result = self.__class__()
result += self
result += other
return result
__radd__ = __add__
def __iadd__(self, other):
if not isinstance(other, Equations):
other = Equations(other, level=1)
self._eq_names = list(set(self._eq_names + other._eq_names)) # what to do if same variables?
self._diffeq_names = list(set(self._diffeq_names + other._diffeq_names))
self._diffeq_names_nonzero = list(set(self._diffeq_names_nonzero + other._diffeq_names_nonzero))
self._function = disjoint_dict_union(self._function, other._function)
self._alias = disjoint_dict_union(self._alias, other._alias)
self._string = disjoint_dict_union(self._string, other._string)
self._namespace = disjoint_dict_union(self._namespace, other._namespace)
# We do this to fix a bug where if you add two Equations together and
# then create groups from them, the add_prefix_namespace step creates
# names which can't be correctly resolved in the second NeuronGroup
# created. This happens because although self._namespace is a new object,
# self._namespace[var] is shared between the two objects.
for var in self._namespace.keys():
self._namespace[var] = copy.copy(self._namespace[var])
try:
self._units = disjoint_dict_union(self._units, other._units)
except AttributeError:
raise DimensionMismatchError("The two sets of equations do not have compatible units")
return self
"""
---------------------------------
OTHER METHODS (CALLED EXTERNALLY)
---------------------------------
"""
def fixed_point(self, **kwd):
'''
Returns a fixed point of the differential equations
as a dictionary. The keyword arguments give the (optional)
initial point (default = 0).
'''
values = {}
for name, value in self._units.iteritems():
values[name] = 0 * value
values.update(kwd)
# Initial vector
x0 = [values[name] for name in self._diffeq_names_nonzero]
# Vector function
def f(x):
# Put the units back
x = [xi * get_unit(x0i) for xi, x0i in zip(x, x0)]
values.update(zip(self._diffeq_names_nonzero, x))
return [self.apply(name, values) for name in self._diffeq_names_nonzero]
xf, _, ier, _ = optimize.fsolve(f, x0, full_output=True)
if ier:
# Put the units back
xf = [xfi * get_unit(x0i) for xfi, x0i in zip(xf, x0)]
# Return a dictionary
return dict(zip(self._diffeq_names_nonzero, xf))
else: # Not found
warnings.warn('Could not find a fixed point of the equations')
return kwd
def substitute(self, name1, name2):
"""
Changes name1 to name2 (variable names).
Note: I don't where this is called!
"""
# Aliases
if name1 in self._alias:
self._alias[name2] = self._alias[name1]
del self._alias[name1]
# Units
if name1 in self._units:
self._units[name2] = self._units[name1]
del self._units[name1]
# Equations
if name1 in self._eq_names:
self._eq_names[self._eq_names.index(name1)] = name2
# Differential equations
if name1 in self._diffeq_names:
self._diffeq_names[self._diffeq_names.index(name1)] = name2
if name1 in self._diffeq_names_nonzero:
self._diffeq_names_nonzero[self._diffeq_names_nonzero.index(name1)] = name2
# Strings
if name1 in self._string:
self._string[name2] = self._string[name1]
del self._string[name1]
for name, value in self._string.iteritems():
self._string[name] = re.sub("\\b" + name1 + "\\b", name2, value)
# Namespaces
if name1 in self._namespace:
self._namespace[name2] = self._namespace[name1]
del self._namespace[name1]
def __getattr__(self, name):
'''
Returns the corresponding function.
'''
# bug with clustertools
if name == 'as_array':
raise Exception()
return lambda ** kwd:self.apply(name, kwd)
def __len__(self):
'''
Number of differential equations
Note: is this still used?
'''
return len(self._diffeq_names)
def __repr__(self):
s = ''
for var in self._diffeq_names:
s += 'd' + var + '/dt = ' + self._string[var] + ' [diffeq]\n'
for var in self._eq_names:
if var in self._alias:
typename = ' [alias]'
else:
typename = ' [eq]'
s += var + ' = ' + self._string[var] + typename + '\n'
return s
def __reduce__(self):
# To avoid recursion, we temporarily set the class to a trivial
# class, this is restored at the end, and by _load_Equations_from_pickle
# too
self.__class__, cls = PickledEquations, self.__class__
selfcopy = copy.copy(self)
# We need to delete __builtins__ from all the namespaces as it is
# not picklable, so we make copies
selfcopy._namespace = copy.copy(selfcopy._namespace)
for key in selfcopy._namespace.keys():
selfcopy._namespace[key] = copy.copy(selfcopy._namespace[key])
if '__builtins__' in selfcopy._namespace[key]:
del selfcopy._namespace[key]['__builtins__']
selfcopy._function = {}
# Sometimes namespaces have numpy ufuncs in them, which are not
# picklable, so we replace them with PickledUfunc objects which just
# store their name, and _load_equations_from_pickle will extract them
# from numpy again
def replaceufunc(d):
for k in d.keys():
v = d[k]
if isinstance(v, numpy.ufunc):
d[k] = PickledUfunc(v)
if isinstance(v, dict):
replaceufunc(v)
replaceufunc(selfcopy.__dict__)
self.__class__ = cls
return (_load_Equations_from_pickle, (selfcopy, cls))
class PickledUfunc(object):
def __init__(self, ufunc):
self.name = ufunc.__name__
def get(self):
return getattr(numpy, self.name)
class PickledEquations(object):
pass
def _load_Equations_from_pickle(eqs, cls):
def replaceufunc(d):
for k in d.keys():
v = d[k]
if isinstance(v, PickledUfunc):
d[k] = v.get()
if isinstance(v, dict):
replaceufunc(v)
replaceufunc(eqs.__dict__)
eqs.__class__ = cls
eqs.prepare()
return eqs
# Utilitary functions
# -------------------
def call_with_dict(f, d):
'''
Calls a function f with arguments from dictionary d.
The dictionary can contain keys that are not variables of f.
'''
return f(*[d[var] for var in f.func_code.co_varnames])
def disjoint_dict_union(d1, d2):
'''
Merges the dictionaries d1 and d2 and checks that
they are compatible (i.e., raises an exception if d1[key]!=d2[key])
'''
result = {}
result.update(d1)
for key, value in d2.iteritems(): # Bug here
if (key in d1) and (d1[key] != value):
raise AttributeError, "Incompatible dictionaries in disjoint union, problem with key " + key
result[key] = value
return result
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