'''
Compartmental models.
This module defines the `SpatialNeuron` class, which defines multicompartmental
models.
'''
import weakref
import copy
import sympy as sp
import numpy as np
from brian2.core.variables import Variables
from brian2.equations.equations import (Equations, PARAMETER, SUBEXPRESSION,
DIFFERENTIAL_EQUATION, SingleEquation,
extract_constant_subexpressions)
from brian2.groups.group import Group, CodeRunner, create_runner_codeobj
from brian2.units.allunits import ohm, siemens, amp, meter, volt
from brian2.units.fundamentalunits import (Quantity,
fail_for_dimension_mismatch,
have_same_dimensions,
DimensionMismatchError)
from brian2.units.stdunits import uF, cm
from brian2.parsing.sympytools import sympy_to_str, str_to_sympy
from brian2.utils.logger import get_logger
from brian2.groups.neurongroup import (NeuronGroup, SubexpressionUpdater,
to_start_stop)
from brian2.groups.subgroup import Subgroup
from brian2.equations.codestrings import Expression
__all__ = ['SpatialNeuron']
logger = get_logger(__name__)
[docs]class FlatMorphology(object):
'''
Container object to store the flattened representation of a morphology.
Note that all values are stored as numpy arrays without unit information
(i.e. in base units).
'''
def __init__(self, morphology):
self.n = n = morphology.total_compartments # Total number of compartments
# Per-compartment attributes
self.length = np.zeros(n)
self.distance = np.zeros(n)
self.area = np.zeros(n)
self.diameter = np.zeros(n)
self.volume = np.zeros(n)
self.r_length_1 = np.zeros(n)
self.r_length_2 = np.zeros(n)
self.start_x = np.zeros(n)
self.start_y = np.zeros(n)
self.start_z = np.zeros(n)
self.x = np.zeros(n)
self.y = np.zeros(n)
self.z = np.zeros(n)
self.end_x = np.zeros(n)
self.end_y = np.zeros(n)
self.end_z = np.zeros(n)
self.depth = np.zeros(n, dtype=np.int32)
self.sections = sections = morphology.total_sections
self.end_distance = np.zeros(sections)
# Index of the parent for each section (-1 for the root)
self.morph_parent_i = np.zeros(sections, dtype=np.int32)
# The children indices for each section (list of lists, will be later
# transformed into an array representation)
self.morph_children = []
# each section is child of exactly one parent, this stores the index in
# the parents list of children
self.morph_idxchild = np.zeros(sections, dtype=np.int32)
self.starts = np.zeros(sections, dtype=np.int32)
self.ends = np.zeros(sections, dtype=np.int32)
# recursively fill the data structures
self._sections_without_coordinates = False
self.has_coordinates = False
self._offset = 0
self._section_counter = 0
self._insert_data(morphology)
if self.has_coordinates and self._sections_without_coordinates:
logger.info('The morphology has a mix of sections with and '
'without coordinates. The SpatialNeuron object '
'will store NaN values for the coordinates of '
'the sections that do not specify coordinates. '
'Call generate_coordinates on the morphology '
'before creating the SpatialNeuron object to fill '
'in the missing coordinates.')
# Do not store coordinates for morphologies that don't define them
if not self.has_coordinates:
self.start_x = self.start_y = self.start_z = None
self.x = self.y = self.z = None
self.end_x = self.end_y = self.end_z = None
# Transform the list of list of children into a 2D array (stored as
# 1D) -- note that this wastes space if the number of children per
# section is very different. In practice, this should not be much of a
# problem since most sections have 0, 1, or 2 children (e.g. SWC files
# on neuromorpho.org are all binary trees)
self.morph_children_num = np.array([len(c)
for c in self.morph_children] + [0])
max_children = max(self.morph_children_num)
morph_children = np.zeros((sections+1, max_children), dtype=np.int32)
for idx, section_children in enumerate(self.morph_children):
morph_children[idx, :len(section_children)] = section_children
self.morph_children = morph_children.reshape(-1)
def _insert_data(self, section, parent_idx=-1, depth=0):
n = section.n
start = self._offset
end = self._offset + n
# Compartment attributes
self.depth[start:end] = depth
self.length[start:end] = np.asarray(section.length)
self.distance[start:end] = np.asarray(section.distance)
self.area[start:end] = np.asarray(section.area)
self.diameter[start:end] = np.asarray(section.diameter)
self.volume[start:end] = np.asarray(section.volume)
self.r_length_1[start:end] = np.asarray(section.r_length_1)
self.r_length_2[start:end] = np.asarray(section.r_length_2)
if section.x is None:
self._sections_without_coordinates = True
self.start_x[start:end] = np.ones(n)*np.nan
self.start_y[start:end] = np.ones(n)*np.nan
self.start_z[start:end] = np.ones(n)*np.nan
self.x[start:end] = np.ones(n)*np.nan
self.y[start:end] = np.ones(n)*np.nan
self.z[start:end] = np.ones(n)*np.nan
self.end_x[start:end] = np.ones(n)*np.nan
self.end_y[start:end] = np.ones(n)*np.nan
self.end_z[start:end] = np.ones(n)*np.nan
else:
self.has_coordinates = True
self.start_x[start:end] = np.asarray(section.start_x)
self.start_y[start:end] = np.asarray(section.start_y)
self.start_z[start:end] = np.asarray(section.start_z)
self.x[start:end] = np.asarray(section.x)
self.y[start:end] = np.asarray(section.y)
self.z[start:end] = np.asarray(section.z)
self.end_x[start:end] = np.asarray(section.end_x)
self.end_y[start:end] = np.asarray(section.end_y)
self.end_z[start:end] = np.asarray(section.end_z)
# Section attributes
idx = self._section_counter
# We start counting from 1 for the parent indices, since the index 0
# is used for the (virtual) root compartment
self.morph_parent_i[idx] = parent_idx + 1
self.morph_children.append([])
self.starts[idx] = start
self.ends[idx] = end
# Append ourselves to the children list of our parent
self.morph_idxchild[idx] = len(self.morph_children[parent_idx+1])
self.morph_children[parent_idx + 1].append(idx + 1)
self.end_distance[idx] = section.end_distance
# Recurse down the tree
self._offset += n
self._section_counter += 1
for child in section.children:
self._insert_data(child, parent_idx=idx, depth=depth+1)
[docs]class SpatialNeuron(NeuronGroup):
'''
A single neuron with a morphology and possibly many compartments.
Parameters
----------
morphology : `Morphology`
The morphology of the neuron.
model : (str, `Equations`)
The equations defining the group.
method : (str, function), optional
The numerical integration method. Either a string with the name of a
registered method (e.g. "euler") or a function that receives an
`Equations` object and returns the corresponding abstract code. If no
method is specified, a suitable method will be chosen automatically.
threshold : str, optional
The condition which produces spikes. Should be a single line boolean
expression.
threshold_location : (int, `Morphology`), optional
Compartment where the threshold condition applies, specified as an
integer (compartment index) or a `Morphology` object corresponding to
the compartment (e.g. ``morpho.axon[10*um]``).
If unspecified, the threshold condition applies at all compartments.
Cm : `Quantity`, optional
Specific capacitance in uF/cm**2 (default 0.9). It can be accessed and
modified later as a state variable. In particular, its value can differ
in different compartments.
Ri : `Quantity`, optional
Intracellular resistivity in ohm.cm (default 150). It can be accessed
as a shared state variable, but modified only before the first run.
It is uniform across the neuron.
reset : str, optional
The (possibly multi-line) string with the code to execute on reset.
events : dict, optional
User-defined events in addition to the "spike" event defined by the
``threshold``. Has to be a mapping of strings (the event name) to
strings (the condition) that will be checked.
refractory : {str, `Quantity`}, optional
Either the length of the refractory period (e.g. ``2*ms``), a string
expression that evaluates to the length of the refractory period
after each spike (e.g. ``'(1 + rand())*ms'``), or a string expression
evaluating to a boolean value, given the condition under which the
neuron stays refractory after a spike (e.g. ``'v > -20*mV'``)
namespace : dict, optional
A dictionary mapping identifier names to objects. If not given, the
namespace will be filled in at the time of the call of `Network.run`,
with either the values from the ``namespace`` argument of the
`Network.run` method or from the local context, if no such argument is
given.
dtype : (`dtype`, `dict`), optional
The `numpy.dtype` that will be used to store the values, or a
dictionary specifying the type for variable names. If a value is not
provided for a variable (or no value is provided at all), the preference
setting `core.default_float_dtype` is used.
dt : `Quantity`, optional
The time step to be used for the simulation. Cannot be combined with
the `clock` argument.
clock : `Clock`, optional
The update clock to be used. If neither a clock, nor the `dt` argument
is specified, the `defaultclock` will be used.
order : int, optional
The priority of of this group for operations occurring at the same time
step and in the same scheduling slot. Defaults to 0.
name : str, optional
A unique name for the group, otherwise use ``spatialneuron_0``, etc.
'''
def __init__(self, morphology=None, model=None, threshold=None,
refractory=False, reset=None, events=None,
threshold_location=None,
dt=None, clock=None, order=0, Cm=0.9 * uF / cm ** 2, Ri=150 * ohm * cm,
name='spatialneuron*', dtype=None, namespace=None,
method=('exact', 'exponential_euler', 'rk2', 'heun'),
method_options=None):
# #### Prepare and validate equations
if isinstance(model, str):
model = Equations(model)
if not isinstance(model, Equations):
raise TypeError(('model has to be a string or an Equations '
'object, is "%s" instead.') % type(model))
# Insert the threshold mechanism at the specified location
if threshold_location is not None:
if hasattr(threshold_location,
'_indices'): # assuming this is a method
threshold_location = threshold_location._indices()
# for now, only a single compartment allowed
if len(threshold_location) == 1:
threshold_location = threshold_location[0]
else:
raise AttributeError(('Threshold can only be applied on a '
'single location'))
threshold = '(' + threshold + ') and (i == ' + str(threshold_location) + ')'
# Check flags (we have point currents)
model.check_flags({DIFFERENTIAL_EQUATION: ('point current',),
PARAMETER: ('constant', 'shared', 'linked', 'point current'),
SUBEXPRESSION: ('shared', 'point current',
'constant over dt')})
#: The original equations as specified by the user (i.e. before
#: inserting point-currents into the membrane equation, before adding
#: all the internally used variables and constants, etc.).
self.user_equations = model
# Separate subexpressions depending whether they are considered to be
# constant over a time step or not (this would also be done by the
# NeuronGroup initializer later, but this would give incorrect results
# for the linearity check)
model, constant_over_dt = extract_constant_subexpressions(model)
# Extract membrane equation
if 'Im' in model:
if len(model['Im'].flags):
raise TypeError('Cannot specify any flags for the transmembrane '
'current Im.')
membrane_expr = model['Im'].expr # the membrane equation
else:
raise TypeError('The transmembrane current Im must be defined')
model_equations = []
# Insert point currents in the membrane equation
for eq in model.values():
if eq.varname == 'Im':
continue # ignore -- handled separately
if 'point current' in eq.flags:
fail_for_dimension_mismatch(eq.dim, amp,
"Point current " + eq.varname + " should be in amp")
membrane_expr = Expression(
str(membrane_expr.code) + '+' + eq.varname + '/area')
eq = SingleEquation(eq.type, eq.varname, eq.dim, expr=eq.expr,
flags=list(set(eq.flags)-{'point current'}))
model_equations.append(eq)
model_equations.append(SingleEquation(SUBEXPRESSION, 'Im',
dimensions=(amp/meter**2).dim,
expr=membrane_expr))
model_equations.append(SingleEquation(PARAMETER, 'v', volt.dim))
model = Equations(model_equations)
###### Process model equations (Im) to extract total conductance and the remaining current
# Expand expressions in the membrane equation
for var, expr in model.get_substituted_expressions(include_subexpressions=True):
if var == 'Im':
Im_expr = expr
break
else:
raise AssertionError('Model equations did not contain Im!')
# Differentiate Im with respect to v
Im_sympy_exp = str_to_sympy(Im_expr.code)
v_sympy = sp.Symbol('v', real=True)
diffed = sp.diff(Im_sympy_exp, v_sympy)
unevaled_derivatives = diffed.atoms(sp.Derivative)
if len(unevaled_derivatives):
raise TypeError('Cannot take the derivative of "{Im}" with respect '
'to v.'.format(Im=Im_expr.code))
gtot_str = sympy_to_str(sp.simplify(-diffed))
I0_str = sympy_to_str(sp.simplify(Im_sympy_exp - diffed*v_sympy))
if gtot_str == '0':
gtot_str += '*siemens/meter**2'
if I0_str == '0':
I0_str += '*amp/meter**2'
gtot_str = "gtot__private=" + gtot_str + ": siemens/meter**2"
I0_str = "I0__private=" + I0_str + ": amp/meter**2"
model += Equations(gtot_str + "\n" + I0_str)
# Insert morphology (store a copy)
self.morphology = copy.deepcopy(morphology)
# Flatten the morphology
self.flat_morphology = FlatMorphology(morphology)
# Equations for morphology
# TODO: check whether Cm and Ri are already in the equations
# no: should be shared instead of constant
# yes: should be constant (check)
eqs_constants = Equations("""
length : meter (constant)
distance : meter (constant)
area : meter**2 (constant)
volume : meter**3
Ic : amp/meter**2
diameter : meter (constant)
Cm : farad/meter**2 (constant)
Ri : ohm*meter (constant, shared)
r_length_1 : meter (constant)
r_length_2 : meter (constant)
time_constant = Cm/gtot__private : second
space_constant = (2/pi)**(1.0/3.0) * (area/(1/r_length_1 + 1/r_length_2))**(1.0/6.0) /
(2*(Ri*gtot__private)**(1.0/2.0)) : meter
""")
if self.flat_morphology.has_coordinates:
eqs_constants += Equations('''
x : meter (constant)
y : meter (constant)
z : meter (constant)
''')
NeuronGroup.__init__(self, morphology.total_compartments,
model=model + eqs_constants,
method_options=method_options,
threshold=threshold, refractory=refractory,
reset=reset, events=events,
method=method, dt=dt, clock=clock, order=order,
namespace=namespace, dtype=dtype, name=name)
# Parameters and intermediate variables for solving the cable equations
# Note that some of these variables could have meaningful physical
# units (e.g. _v_star is in volt, _I0_all is in amp/meter**2 etc.) but
# since these variables should never be used in user code, we don't
# assign them any units
self.variables.add_arrays(['_ab_star0', '_ab_star1', '_ab_star2',
'_b_plus', '_b_minus',
'_v_star', '_u_plus', '_u_minus',
'_v_previous', '_c',
# The following two are only necessary for
# C code where we cannot deal with scalars
# and arrays interchangeably:
'_I0_all', '_gtot_all'],
size=self.N, read_only=True)
self.Cm = Cm
self.Ri = Ri
# These explict assignments will load the morphology values from disk
# in standalone mode
self.distance_ = self.flat_morphology.distance
self.length_ = self.flat_morphology.length
self.area_ = self.flat_morphology.area
self.diameter_ = self.flat_morphology.diameter
self.r_length_1_ = self.flat_morphology.r_length_1
self.r_length_2_ = self.flat_morphology.r_length_2
if self.flat_morphology.has_coordinates:
self.x_ = self.flat_morphology.x
self.y_ = self.flat_morphology.y
self.z_ = self.flat_morphology.z
# Performs numerical integration step
self.add_attribute('diffusion_state_updater')
self.diffusion_state_updater = SpatialStateUpdater(self, method,
clock=self.clock,
order=order)
# Update v after the gating variables to obtain consistent Ic and Im
self.diffusion_state_updater.order = 1
# Creation of contained_objects that do the work
self.contained_objects.extend([self.diffusion_state_updater])
if len(constant_over_dt):
self.subexpression_updater = SubexpressionUpdater(self,
constant_over_dt)
self.contained_objects.append(self.subexpression_updater)
def __getattr__(self, name):
'''
Subtrees are accessed by attribute, e.g. neuron.axon.
'''
return self.spatialneuron_attribute(self, name)
def __getitem__(self, item):
'''
Selects a segment, where x is a slice of either compartment
indexes or distances.
Note a: segment is not a SpatialNeuron, only a Group.
'''
return self.spatialneuron_segment(self, item)
@staticmethod
def _find_subtree_end(morpho):
'''
Go down a morphology recursively to find the (absolute) index of the
"final" compartment (i.e. the one with the highest index) of the
subtree.
Parameters
----------
morpho : `Morphology`
The morphology for which to find the index.
Returns
-------
index : int
The highest index within the subtree.
'''
indices = [morpho.indices[-1]]
for child in morpho.children:
indices.append(SpatialNeuron._find_subtree_end(child))
return max(indices)
[docs] @staticmethod
def spatialneuron_attribute(neuron, name):
'''
Selects a subtree from `SpatialNeuron` neuron and returns a `SpatialSubgroup`.
If it does not exist, returns the `Group` attribute.
'''
if name == 'main': # Main section, without the subtrees
indices = neuron.morphology.indices[:]
start, stop = indices[0], indices[-1]
return SpatialSubgroup(neuron, start, stop + 1,
morphology=neuron.morphology)
elif (name != 'morphology') and ((name in getattr(neuron.morphology, 'children', [])) or
all([c in 'LR123456789' for c in name])): # subtree
morpho = neuron.morphology[name]
start = morpho.indices[0]
stop = SpatialNeuron._find_subtree_end(morpho)
return SpatialSubgroup(neuron, start, stop + 1, morphology=morpho)
else:
return Group.__getattr__(neuron, name)
[docs] @staticmethod
def spatialneuron_segment(neuron, item):
'''
Selects a segment from `SpatialNeuron` neuron, where item is a slice of
either compartment indexes or distances.
Note a: segment is not a `SpatialNeuron`, only a `Group`.
'''
if isinstance(item, slice) and isinstance(item.start, Quantity):
if item.step is not None:
raise ValueError('Cannot specify a step size for slicing based'
'on length.')
start, stop = item.start, item.stop
if (not have_same_dimensions(start, meter) or
not have_same_dimensions(stop, meter)):
raise DimensionMismatchError('Start and stop should have units '
'of meter', start, stop)
# Convert to integers (compartment numbers)
indices = neuron.morphology.indices[item]
start, stop = indices[0], indices[-1] + 1
elif not isinstance(item, slice) and hasattr(item, 'indices'):
start, stop = to_start_stop(item.indices[:], neuron._N)
else:
start, stop = to_start_stop(item, neuron._N)
if start >= stop:
raise IndexError('Illegal start/end values for subgroup, %d>=%d' %
(start, stop))
return Subgroup(neuron, start, stop)
[docs]class SpatialSubgroup(Subgroup):
'''
A subgroup of a `SpatialNeuron`.
Parameters
----------
source : int
First compartment.
stop : int
Ending compartment, not included (as in slices).
morphology : `Morphology`
Morphology corresponding to the subgroup (not the full
morphology).
name : str, optional
Name of the subgroup.
'''
def __init__(self, source, start, stop, morphology, name=None):
self.morphology = morphology
Subgroup.__init__(self, source, start, stop, name)
def __getattr__(self, name):
return SpatialNeuron.spatialneuron_attribute(self, name)
def __getitem__(self, item):
return SpatialNeuron.spatialneuron_segment(self, item)
[docs]class SpatialStateUpdater(CodeRunner, Group):
'''
The `CodeRunner` that updates the state variables of a `SpatialNeuron`
at every timestep.
'''
def __init__(self, group, method, clock, order=0):
# group is the neuron (a group of compartments)
self.method_choice = method
self.group = weakref.proxy(group)
compartments = group.flat_morphology.n
sections = group.flat_morphology.sections
CodeRunner.__init__(self, group,
'spatialstateupdate',
code='''_gtot = gtot__private
_I0 = I0__private''',
clock=clock,
when='groups',
order=order,
name=group.name + '_spatialstateupdater*',
check_units=False,
template_kwds={'number_sections': sections})
self.variables = Variables(self, default_index='_section_idx')
self.variables.add_reference('N', group)
# One value per compartment
self.variables.add_arange('_compartment_idx', size=compartments)
self.variables.add_array('_invr', dimensions=siemens.dim,
size=compartments, constant=True,
index='_compartment_idx')
# one value per section
self.variables.add_arange('_section_idx', size=sections)
self.variables.add_array('_P_parent', size=sections,
constant=True) # elements below diagonal
self.variables.add_arrays(['_morph_idxchild', '_morph_parent_i',
'_starts', '_ends'], size=sections,
dtype=np.int32, constant=True)
self.variables.add_arrays(['_invr0', '_invrn'], dimensions=siemens.dim,
size=sections, constant=True)
# one value per section + 1 value for the root
self.variables.add_arange('_section_root_idx', size=sections+1)
self.variables.add_array('_P_diag', size=sections+1,
constant=True, index='_section_root_idx')
self.variables.add_array('_B', size=sections+1,
constant=True, index='_section_root_idx')
self.variables.add_array('_morph_children_num',
size=sections+1, dtype=np.int32,
constant=True, index='_section_root_idx')
# 2D matrices of size (sections + 1) x max children per section
self.variables.add_arange('_morph_children_idx',
size=len(group.flat_morphology.morph_children))
self.variables.add_array('_P_children',
size=len(group.flat_morphology.morph_children),
index='_morph_children_idx',
constant=True) # elements above diagonal
self.variables.add_array('_morph_children',
size=len(group.flat_morphology.morph_children),
dtype=np.int32, constant=True,
index='_morph_children_idx')
self._enable_group_attributes()
self._morph_parent_i = group.flat_morphology.morph_parent_i
self._morph_children_num = group.flat_morphology.morph_children_num
self._morph_children = group.flat_morphology.morph_children
self._morph_idxchild = group.flat_morphology.morph_idxchild
self._starts = group.flat_morphology.starts
self._ends = group.flat_morphology.ends
[docs] def before_run(self, run_namespace):
super(SpatialStateUpdater, self).before_run(run_namespace)
# Raise a warning if the slow pure numpy version is used
from brian2.codegen.runtime.numpy_rt.numpy_rt import NumpyCodeObject
if type(self.code_objects[0]) == NumpyCodeObject:
# If numpy is used, raise a warning if scipy is not present
try:
import scipy
except ImportError:
logger.info(('SpatialNeuron will use numpy to do the numerical '
'integration -- this will be very slow. Either '
'switch to a different code generation target '
'(e.g. cython) or install scipy.'),
once=True)