diff --git a/doc/conf.py b/doc/conf.py
index 0ac043e2704231874e3c9e743b0567bed0f4cd23..79f75ff34ec2fe9df4eb0c2bd1e9970237b50136 100644
--- a/doc/conf.py
+++ b/doc/conf.py
@@ -17,6 +17,7 @@ def setup(app):
app.add_js_file("latest-warning.js")
extensions = [
+ 'sphinx.ext.intersphinx',
'sphinx.ext.autodoc',
'sphinx.ext.todo',
'sphinx.ext.mathjax',
@@ -28,6 +29,10 @@ master_doc = 'index'
html_logo = 'images/arbor-lines-proto-colour.svg'
html_favicon = 'images/arbor-lines-proto-colour-notext.svg'
+intersphinx_mapping = {
+ 'sphinx': ('https://lfpykit.readthedocs.io/en/latest/', None),
+}
+
project = 'Arbor'
copyright = '2017-2021, ETHZ & FZJ'
author = 'ETHZ & FZJ'
diff --git a/doc/scripts/make_images.py b/doc/scripts/make_images.py
index 9bfa9511a35f0678fadd523650049a2816ab8d6f..35369ec54205f50ce7287907d729964f6a8dcd2a 100644
--- a/doc/scripts/make_images.py
+++ b/doc/scripts/make_images.py
@@ -311,6 +311,7 @@ def generate(path=''):
####################### Tutorial examples
morph_image([inputs.tutorial_morph], ['segments'], path+'/tutorial_morph.svg')
+ morph_image([inputs.tutorial_morph], ['segments'], path+'/tutorial_morph_nolabels_nocolors.svg', colors=False, draw_labels=False)
morph_image([inputs.tutorial_network_ring_morph], ['segments'], path+'/tutorial_network_ring_morph.svg',lab_sc=6)
####################### locsets
diff --git a/doc/tutorial/index.rst b/doc/tutorial/index.rst
index b5a897deea6baf00fe4fe0fa58e869bb2f894219..9d57b4d7db0910e8a5838b7aa4a5a32403c70c5b 100644
--- a/doc/tutorial/index.rst
+++ b/doc/tutorial/index.rst
@@ -7,7 +7,7 @@ Tutorials
You can find some examples of full Arbor simulations in the ``python/examples`` directory of the
`Arbor repository <https://github.com/arbor-sim/arbor>`_.
- The examples use ``pandas`` and ``seaborn`` for analysis and plotting which are expected to be
+ The examples use ``pandas``, ``seaborn`` and ``LFPykit`` for analysis and plotting which are expected to be
installed independently from Arbor.
In an interactive Python interpreter, you can use ``help()`` on any class or function to get its
@@ -28,3 +28,4 @@ Tutorials
network_ring
network_ring_mpi
two_cells_gap_junctions
+ tutorial_lfpykit
diff --git a/doc/tutorial/tutorial_lfpykit.rst b/doc/tutorial/tutorial_lfpykit.rst
new file mode 100644
index 0000000000000000000000000000000000000000..31b1080efb59912c299dc85abc0689cc0251be5a
--- /dev/null
+++ b/doc/tutorial/tutorial_lfpykit.rst
@@ -0,0 +1,227 @@
+.. _tutorial_lfpykit:
+
+Extracellular signals
+=====================
+
+This example takes elements from other tutorials to create a geometrically
+detailed single cell model from an SWC morphology file, and adds predictions of
+extracellular potentials using the `LFPykit <https://lfpykit.readthedocs.io/en/latest>`_ Python library.
+LFPykit provides a few different classes facilitating
+calculations of extracellular potentials and related electroencephalography (EEG)
+and magnetoencephalography (MEG) signals from geometrically detailed neuron models under various assumptions.
+These are signals that mainly stem from transmembrane currents.
+
+.. Note::
+
+ **Concepts covered in this tutorial:**
+
+ 1. Recording of transmembrane currents using :class:`arbor.cable_probe_total_current_cell`
+ 2. Recording of stimulus currents using :class:`arbor.cable_probe_stimulus_current_cell`
+ 3. Using the :class:`arbor.place_pwlin` API
+ 4. Map recorded transmembrane currents to extracellular potentials by derivinging Arbor specific classes from LFPykit's :class:`lfpykit.LineSourcePotential` and :class:`lfpykit.CellGeometry`
+
+.. _tutorial_lfpykit-linesource:
+
+The line source approximation
+-----------------------------
+
+First, let's describe how one can compute extracellular potentials from transmembrane currents of a number of segments,
+assuming that each segment can be treated as an equivalent line current source using a formalism invented
+by Gary R. Holt and Christof Koch [1]_.
+The implementation used in this tutorial rely on :class:`lfpykit.LineSourcePotential`.
+This class conveniently defines a 2D linear response matrix
+:math:`\mathbf{M}` between transmembrane current array
+:math:`\mathbf{I}` (nA)Â of a neuron model's
+geometry (:class:`lfpykit.CellGeometry`) and the
+corresponding extracellular electric potential in different extracellular locations
+:math:`\mathbf{V}_{e}` (mV) so
+
+.. math:: \mathbf{V}_{e} = \mathbf{M} \mathbf{I}
+
+The transmembrane current :math:`\mathbf{I}` is an array of shape (# segments, # timesteps)
+with unit (nA), and each row indexed by :math:`j` of
+:math:`\mathbf{V}_{e}` represents the electric potential at each
+measurement site for every time step.
+The resulting shape of :math:`\mathbf{V}_{e}` is (# locations, # timesteps).
+The elements of :math:`\mathbf{M}` are computed as:
+
+.. math:: M_{ji} = \frac{1}{ 4 \pi \sigma L_i } \log
+ \left|
+ \frac{\sqrt{h_{ji}^2+r_{ji}^2}-h_{ji}
+ }{
+ \sqrt{l_{ji}^2+r_{ji}^2}-l_{ji}}
+ \right|~.
+
+
+Here, segments are indexed by :math:`i`,
+segment length is denoted :math:`L_i`, perpendicular distance from the
+electrode point contact to the axis of the line segment is denoted
+:math:`r_{ji}`, longitudinal distance measured from the start of the
+segment is denoted :math:`h_{ji}` and longitudinal distance from the other
+end of the segment is denoted :math:`l_{ji}= L_i + h_{ji}`.
+
+.. Note::
+
+ **Assumptions:**
+
+ 1. The extracellular conductivity :math:`\sigma` is infinite, homogeneous, frequency independent (linear) and isotropic
+ 2. Each segment is treated as a straight line source with homogeneous current density between its start and end point coordinate.
+ Although Arbor allows segments to be defined as conical frusta with varying radius, we shall assume that any variation in
+ radius is small relative to overall segment length.
+ 3. Each extracellular measurement site is treated as a point
+ 4. The minimum distance to a line source is set equal to the average segment radius to avoid singularities.
+
+
+.. _tutorial_lfpykit-model:
+
+The model
+---------
+
+In this tutorial, the neuron model itself is kept deliberately simple with only
+passive (leaky) membrane dynamics, and it receives sinusoid synaptic current
+input in one arbitrary chosen control volume (CV).
+
+First we import some required modules:
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 14-17
+
+We define a basic :class:`Recipe <arbor.recipe>` class, holding a cell and three probes (voltage, stimulus current and total current):
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 22-54
+
+Then, load a morphology on ``SWC`` file format (interpreted according to :ref:`Arbor's specifications <morph-formats>`).
+Similar to the tutorial :ref:`"A simple single cell model" <tutorialsinglecellswc>`,
+we use the morphology file in ``python/example/single_cell_detailed.swc``:
+
+.. figure:: ../gen-images/tutorial_morph_nolabels_nocolors.svg
+ :width: 800
+ :align: left
+
+ A graphic depiction of ``single_cell_detailed.swc``.
+
+Pass the filename as an argument to the simulation script:
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 57-66
+
+As a target for a current stimuli we define an :class:`arbor.location` :
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 68-69
+
+Next, we let a basic function define our cell model, taking the morphology and clamp location as
+input.
+The function defines various attributes (:class:`~arbor.label_dict`, :class:`~arbor.decor`) for
+the cell model,
+sets sinusoid current clamp as stimuli using :class:`~arbor.iclamp`
+defines discretization policy (:class:`~arbor.cv_policy_fixed_per_branch`)
+and returns the corresponding :class:`~arbor.place_pwlin` and :class:`~arbor.cable_cell` objects for use later:
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 72-115
+
+Store the function output:
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 119
+
+Next, we instantiate :class:`Recipe`, a :class:`~arbor.context` etc. and execute cell model for a few hundred ms,
+sampling the different signals every 1 ms:
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 121-136
+
+Extract recorded membrane voltages, electrode and transmembrane currents.
+Note that membrane voltages at branch points and intersections between CVs are dropped as
+we only illustrate membrane voltages of segments with finite lengths.
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 138-153
+
+Finally we sum the stimulation and transmembrane currents, allowing the stimuli to mimic a synapse
+current embedded in the membrane itself rather than an intracellular electrode current:
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 161
+
+.. _tutorial_lfpykit-lfpykit:
+
+Compute extracellular potentials
+--------------------------------
+
+Here we utilize the LFPykit library to map
+transmembrane currents recorded during the
+simulation to extracellular potentials in vicinity to the cell.
+We shall account for every segment in each CV using the so-called line-source approximation described :ref:`above <tutorial_lfpykit-linesource>`.
+
+First we define a couple of inherited classes to interface LFPykit
+(as this library is not solely written for Arbor).
+Starting with a class inherited from :class:`lfpykit.CellGeometry`:
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 171-204
+
+Then, a class inherited from :class:`lfpykit.LineSourcePotential`.
+Other use cases may inherit from any other parent class defined in :mod:`lfpykit.models` in a similar manner:
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 207-254
+
+With these two classes one may then compute extracellular potentials from transmembrane
+currents in space with a few lines of code:
+
+.. literalinclude:: ../../python/example/single_cell_extracellular_potentials.py
+ :language: python
+ :lines: 257-280
+
+.. _tutorial_lfpykit-illustration:
+
+The result
+----------
+
+The visualization below of simulation results shows the cellular geometry and a contour plot
+of the extracellular potential (``V_e``) in a plane.
+Each part (CV) of the cell is shown with some color coding for the membrane potential (``V_m``).
+The stimulus location is denoted by the black marker.
+
+.. figure:: tutorial_lfpykit.svg
+ :width: 1600
+ :align: center
+
+ The result.
+
+.. Note::
+
+ The spatial discretization is here deliberately coarse with only 3 CVs per branch.
+ Hence the branch receiving input about 1/6 of the way from its root
+ (from ``decor.place(str(arbor.location(4, 1/6)), iclamp, '"iclamp"')``) is treated as 3 separate
+ line sources with inhomogeneous current density per unit length. This inhomogeneity
+ is due to the fact that the total transmembrane current per CV may
+ be distributed across multiple segments with varying surface area. The transmembrane
+ current is assumed to be constant per unit length per segment.
+
+.. _tutorial_lfpykit-code:
+
+The full code
+-------------
+You can find the full code of the example at ``python/examples/single_cell_extracellular_potentials.py``.
+
+Find the source of LFPykit `here <https://github.com/LFPy/LFPykit/>`_.
+
+References
+----------
+.. [1] Holt, G.R., Koch, C. Electrical Interactions via the Extracellular Potential Near Cell Bodies.
+ J Comput Neurosci 6, 169–184 (1999). https://doi.org/10.1023/A:1008832702585
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diff --git a/python/example/single_cell_extracellular_potentials.py b/python/example/single_cell_extracellular_potentials.py
index adf94ed3c8baa101ae4ad1198c1d3dbf66773c5e..318d7b117dd2f9f95d5dd9a148cf6da7675f1ce5 100644
--- a/python/example/single_cell_extracellular_potentials.py
+++ b/python/example/single_cell_extracellular_potentials.py
@@ -1,5 +1,4 @@
#!/usr/bin/env python
-# coding: utf-8
# Example utilizing the **`LFPykit`** module (https://lfpykit.readthedocs.io,
# https://github.com/LFPy/LFPykit) for predictions of extracellular
@@ -17,12 +16,10 @@ import numpy as np
import arbor
import lfpykit
import matplotlib.pyplot as plt
-from matplotlib.gridspec import GridSpec
from matplotlib.collections import PolyCollection
-import pandas as pd
-class Recipe (arbor.recipe):
+class Recipe(arbor.recipe):
def __init__(self, cell):
super().__init__()
@@ -65,54 +62,61 @@ if len(sys.argv) < 2:
filename = sys.argv[1]
-# define morphology (needed for arbor.place_pwlin)
+# define morphology (needed for ``arbor.place_pwlin`` and ``arbor.cable_cell`` below)
morphology = arbor.load_swc_arbor(filename)
-# number of CVs per branch
-nseg = 3
-
-# Label dictionary
-defs = {}
-labels = arbor.label_dict(defs)
-
-# decor
-decor = arbor.decor()
-
-# set initial voltage, temperature, axial resistivity, membrane capacitance
-decor.set_property(
- Vm=-65, # Initial membrane voltage [mV]
- tempK=300, # Temperature [Kelvin]
- rL=10000, # Axial resistivity [Ω cm]
- cm=0.01, # Membrane capacitance [F/m**2]
-)
-
-# set passive mechanism all over
-# passive mech w. leak reversal potential (mV)
-pas = arbor.mechanism('pas/e=-65')
-pas.set('g', 0.0001) # leak conductivity (S/cm2)
-decor.paint('(all)', arbor.density(pas))
-
-# set sinusoid input current at mid point of terminating CV (segment)
-iclamp = arbor.iclamp(5, # stimulation onset (ms)
- 1E8, # stimulation duration (ms)
- -0.001, # stimulation amplitude (nA)
- frequency=0.1, # stimulation frequency (kHz)
- phase=0) # stimulation phase)
-try:
- # arbor >= 0.5.2 fix
- decor.place('(location 4 0.16667)', iclamp, '"iclamp"')
-except TypeError:
- decor.place('(location 4 0.16667)', iclamp)
-
-# number of CVs per branch
-policy = arbor.cv_policy_fixed_per_branch(nseg)
-decor.discretization(policy)
-
-# create cell and set properties
-cell = arbor.cable_cell(morphology, labels, decor)
-
-# create single cell model
-model = arbor.single_cell_model(cell)
+# define a location on morphology for current clamp
+clamp_location = arbor.location(4, 1/6)
+
+
+def make_cable_cell(morphology, clamp_location):
+ # number of CVs per branch
+ cvs_per_branch = 3
+
+ # Label dictionary
+ defs = {}
+ labels = arbor.label_dict(defs)
+
+ # decor
+ decor = arbor.decor()
+
+ # set initial voltage, temperature, axial resistivity, membrane capacitance
+ decor.set_property(
+ Vm=-65, # Initial membrane voltage (mV)
+ tempK=300, # Temperature (Kelvin)
+ rL=10000, # Axial resistivity (Ω cm)
+ cm=0.01, # Membrane capacitance (F/m**2)
+ )
+
+ # set passive mechanism all over
+ # passive mech w. leak reversal potential (mV)
+ pas = arbor.mechanism('pas/e=-65')
+ pas.set('g', 0.0001) # leak conductivity (S/cm2)
+ decor.paint('(all)', arbor.density(pas))
+
+ # set number of CVs per branch
+ policy = arbor.cv_policy_fixed_per_branch(cvs_per_branch)
+ decor.discretization(policy)
+
+ # place sinusoid input current
+ iclamp = arbor.iclamp(5, # stimulation onset (ms)
+ 1E8, # stimulation duration (ms)
+ -0.001, # stimulation amplitude (nA)
+ frequency=0.1, # stimulation frequency (kHz)
+ phase=0) # stimulation phase)
+ decor.place(str(clamp_location), iclamp, '"iclamp"')
+
+ # create ``arbor.place_pwlin`` object
+ p = arbor.place_pwlin(morphology)
+
+ # create cell and set properties
+ cell = arbor.cable_cell(morphology, labels, decor)
+
+ return p, cell
+
+
+# get place_pwlin and cable_cell objects
+p, cell = make_cable_cell(morphology, clamp_location)
# instantiate recipe with cell
recipe = Recipe(cell)
@@ -122,8 +126,8 @@ context = arbor.context()
domains = arbor.partition_load_balance(recipe, context)
sim = arbor.simulation(recipe, domains, context)
-# set up sampling on probes
-schedule = arbor.regular_schedule(0.1)
+# set up sampling on probes with sampling every 1 ms
+schedule = arbor.regular_schedule(1.)
v_handle = sim.sample(recipe.vprobe_id, schedule, arbor.sampling_policy.exact)
i_handle = sim.sample(recipe.iprobe_id, schedule, arbor.sampling_policy.exact)
c_handle = sim.sample(recipe.cprobe_id, schedule, arbor.sampling_policy.exact)
@@ -131,91 +135,156 @@ c_handle = sim.sample(recipe.cprobe_id, schedule, arbor.sampling_policy.exact)
# run simulation for 500 ms of simulated activity and collect results.
sim.run(tfinal=500)
-# extract time, V_m and I_m for each compartment
+# extract time, V_m, I_m and I_c for each CV
V_m_samples, V_m_meta = sim.samples(v_handle)[0]
I_m_samples, I_m_meta = sim.samples(i_handle)[0]
I_c_samples, I_c_meta = sim.samples(c_handle)[0]
-
-# drop recorded V_m values and corresponding meta data of
-# zero-sized CVs (branch-point potentials)
+# drop recorded V_m values and corresponding metadata of
+# zero-sized CVs (branch-point potentials).
+# Here this is done as the V_m values are only used for visualization purposes
inds = np.array([m.dist != m.prox for m in V_m_meta])
V_m_samples = V_m_samples[:, np.r_[True, inds]]
V_m_meta = np.array(V_m_meta)[inds].tolist()
-
-# note: the cables comprising the metadata for each probe
-# should be the same, as well as the reported sample times.
+# assert that the remaining cables comprising the metadata for each probe
+# are identical, as well as the reported sample times.
assert V_m_meta == I_m_meta
assert (V_m_samples[:, 0] == I_m_samples[:, 0]).all()
-# prep recorded data for plotting
+# prep recorded data for plotting and computation of extracellular potentials
time = V_m_samples[:, 0]
-V_m = V_m_samples[:, 1:].T
-I_m = I_m_samples[:, 1:].T
-I_c = I_c_samples[:, 1:].T
-
+V_m = V_m_samples[:, 1:]
-# gather geometry of CVs and assign segments to each CV
-p = arbor.place_pwlin(morphology)
-x, y, z, d = [np.array([], dtype=float).reshape((0, 2))] * 4
-CV_ind = np.array([], dtype=int) # tracks which CV owns segment
-for i, m in enumerate(I_m_meta):
- segs = p.segments([m])
- for j, seg in enumerate(segs):
- x = np.row_stack([x, [seg.prox.x, seg.dist.x]])
- y = np.row_stack([y, [seg.prox.y, seg.dist.y]])
- z = np.row_stack([z, [seg.prox.z, seg.dist.z]])
- d = np.row_stack([d, [seg.prox.radius * 2, seg.dist.radius * 2]])
- CV_ind = np.r_[CV_ind, i]
+# Add stimulation current to transmembrane current to mimic sinusoid synapse
+# current embedded in the membrane.
+I_m = I_c_samples[:, 1:] + I_m_samples[:, 1:] # (nA)
###############################################################################
-# compute extracellular potential using segment information
+# Compute extracellular potentials
###############################################################################
-cell_geometry = lfpykit.CellGeometry(
- x=x,
- y=y,
- z=z,
- d=d
-)
-
-# membrane voltages, transmemrbane current and corresponding times
-cell_geometry.V_m = V_m # mV
-# nA, sum stimulation and transmembrane current to mimic sinusoid synapse
-cell_geometry.I_m = I_m + I_c
-cell_geometry.time = time # ms
-
-# locations where extracellular potential is predicted
-dx = 1
-dz = 1
-axis = np.round([x.min() - 10, x.max() + 10, y.min() - 10, y.max() + 10])
-# axis = np.round(axis)
-X, Y = np.meshgrid(np.linspace(axis[0], axis[1], int(np.diff(axis[:2]) // dx) + 1),
- np.linspace(axis[2], axis[3], int(np.diff(axis[2:]) // dz) + 1))
-Z = np.zeros_like(X)
-# LineSourcePotential object, get mapping for all segments per CV
-lsp = lfpykit.LineSourcePotential(cell=cell_geometry,
- x=X.flatten(),
- y=Y.flatten(),
- z=Z.flatten())
-M_tmp = lsp.get_transformation_matrix()
+# ## Compute extracellular potentials
+# First we define a couple of classes to interface the LFPykit
+# library (https://LFPykit.readthedocs.io, https://github.com/LFPy/LFPykit):
+class ArborCellGeometry(lfpykit.CellGeometry):
+ '''
+ Class inherited from ``lfpykit.CellGeometry`` for easier forward-model
+ predictions in Arbor that keeps track of arbor.segment information
+ for each CV.
+
+ Parameters
+ ----------
+ p: ``arbor.place_pwlin`` object
+ 3-d locations and cables in a morphology (cf. ``arbor.place_pwlin``)
+ cables: ``list``
+ ``list`` of corresponding ``arbor.cable`` objects where transmembrane
+ currents are recorded (cf. ``arbor.cable_probe_total_current_cell``)
+
+ See also
+ --------
+ lfpykit.CellGeometry
+ '''
+
+ def __init__(self, p, cables):
+ x, y, z, d = [np.array([], dtype=float).reshape((0, 2))] * 4
+ CV_ind = np.array([], dtype=int) # tracks which CV owns segment
+ for i, m in enumerate(cables):
+ segs = p.segments([m])
+ for j, seg in enumerate(segs):
+ x = np.row_stack([x, [seg.prox.x, seg.dist.x]])
+ y = np.row_stack([y, [seg.prox.y, seg.dist.y]])
+ z = np.row_stack([z, [seg.prox.z, seg.dist.z]])
+ d = np.row_stack(
+ [d, [seg.prox.radius * 2, seg.dist.radius * 2]])
+ CV_ind = np.r_[CV_ind, i]
+
+ super().__init__(x=x, y=y, z=z, d=d)
+ self._CV_ind = CV_ind
+
+
+class ArborLineSourcePotential(lfpykit.LineSourcePotential):
+ '''subclass of ``lfpykit.LineSourcePotential`` modified for
+ instances of ``ArborCellGeometry``.
+ Each CV may consist of several segments , and this implementation
+ accounts for their contributions normalized by surface area, that is,
+ we assume constant transmembrane current density per area across each CV
+ and constant current source density per unit length per segment
+ (inherent in the line-source approximation).
+
+ Parameters
+ ----------
+ cell: object
+ ``ArborCellGeometry`` instance or similar.
+ x: ndarray of floats
+ x-position of measurement sites (µm)
+ y: ndarray of floats
+ y-position of measurement sites (µm)
+ z: ndarray of floats
+ z-position of measurement sites (µm)
+ sigma: float > 0
+ scalar extracellular conductivity (S/m)
+
+ See also
+ --------
+ lfpykit.LineSourcePotential
+ '''
+
+ def __init__(self, **kwargs):
+ super().__init__(**kwargs)
+ self._get_transformation_matrix = super().get_transformation_matrix
+
+ def get_transformation_matrix(self):
+ '''Get linear response matrix
+
+ Returns
+ -------
+ response_matrix: ndarray
+ shape (n_coords, n_CVs) ndarray
+ '''
+ M_tmp = self._get_transformation_matrix()
+ n_CVs = np.unique(self.cell._CV_ind).size
+ M = np.zeros((self.x.size, n_CVs))
+ for i in range(n_CVs):
+ inds = self.cell._CV_ind == i
+ M[:, i] = M_tmp[:, inds] @ (self.cell.area[inds] /
+ self.cell.area[inds].sum())
+
+ return M
+
+
+# create ``ArborCellGeometry`` instance
+cell_geometry = ArborCellGeometry(p, I_m_meta)
+
+# define locations where extracellular potential is predicted in vicinity
+# of cell.
+# Axis limits [x-min, x-max, y-min, y-max] (µm)
+axis = np.array([-110, 370, -80, 70])
+dx = 2 # spatial resolution along x-axis (µm)
+dz = 2 # spatial resolution along y-axis (µm)
+X, Y = np.meshgrid(np.linspace(axis[0], axis[1],
+ int(np.diff(axis[:2]) // dx) + 1),
+ np.linspace(axis[2], axis[3],
+ int(np.diff(axis[2:]) // dz) + 1))
+Z = np.zeros_like(X)
-# Define response matrix from M with columns weighted by area of each frusta
-M = np.zeros((lsp.x.size, I_m.shape[0]))
-for i in range(I_m.shape[0]):
- inds = CV_ind == i
- M[:, i] = M_tmp[:, inds] @ (cell_geometry.area[inds] /
- cell_geometry.area[inds].sum())
+# ``ArborLineSourcePotential`` instance, get mapping for all segments per CV
+lsp = ArborLineSourcePotential(cell=cell_geometry,
+ x=X.flatten(),
+ y=Y.flatten(),
+ z=Z.flatten())
+M = lsp.get_transformation_matrix()
-# Extracellular potential using segment information at last time step
-# in x,y-plane coordinates
-V_e = M @ cell_geometry.I_m[:, -1]
+# Extracellular potential in x,y-plane (mV)
+V_e = M @ I_m.T
-# ## Plotting
-# Plot the morphology and extracellular potential prediction
+###############################################################################
+# Plotting
+###############################################################################
+# Plot the morphology and extracellular potential prediction.
+# First we define a couple helper functions:
def create_polygon(x, y, d):
"""create a outline for each segment defined by 1D arrays `x`, `y`, `d`
in x,y-plane which can be drawn using `plt.Polygon`
@@ -272,6 +341,60 @@ def create_polygon(x, y, d):
return list(zip(x, y))
+def get_cv_polycollection(
+ cell_geometry, V_m, vlims=[-66, -64], cmap='viridis'):
+ '''
+ Parameters
+ ----------
+ cell_geometry: ``ArborCellGeometry`` object
+ V_m: ndarray
+ membrane voltages at some time point
+ vlims: list
+ color limits
+ cmap: str
+ matplotlib colormap name
+
+ Returns
+ -------
+ PolyCollection
+ '''
+ norm = plt.Normalize(vmin=vlims[0], vmax=vlims[1], clip=True)
+ colors = [plt.get_cmap(cmap)(norm(v)) for v in V_m]
+ zips = []
+ for i in range(V_m.size):
+ inds = cell_geometry._CV_ind == i
+ zips.append(create_polygon(cell_geometry.x[inds, ].flatten(),
+ cell_geometry.y[inds, ].flatten(),
+ cell_geometry.d[inds, ].flatten()))
+ polycol = PolyCollection(zips,
+ edgecolors=colors,
+ facecolors=colors,
+ linewidths=0.)
+ return polycol
+
+
+def get_segment_outlines(cell_geometry):
+ '''
+ Parameters
+ ----------
+ cell_geometry: ``ArborCellGeometry`` object
+ cmap: str
+ matplotlib colormap name
+
+ Returns
+ -------
+ PolyCollection
+ '''
+ zips = []
+ for x_, y_, d_ in zip(cell_geometry.x, cell_geometry.y, cell_geometry.d):
+ zips.append(create_polygon(x_, y_, d_))
+ polycol = PolyCollection(zips,
+ edgecolors='k',
+ facecolors='none',
+ linewidths=0.5)
+ return polycol
+
+
def colorbar(fig, ax, im,
width=0.01,
height=1.0,
@@ -279,7 +402,7 @@ def colorbar(fig, ax, im,
voffset=0.0,
orientation='vertical'):
'''
- draw matplotlib colorbar without resizing the axes object
+ draw matplotlib colorbar without resizing the parent axes object
'''
rect = np.array(ax.get_position().bounds)
rect = np.array(ax.get_position().bounds)
@@ -289,54 +412,69 @@ def colorbar(fig, ax, im,
caxrect[2] = rect[2] * width
caxrect[3] = rect[3] * height
cax = fig.add_axes(caxrect)
+ cax.grid(False)
cb = fig.colorbar(im, cax=cax, orientation=orientation)
return cb
-fig, ax = plt.subplots(1, 1, figsize=(12, 4))
+# show predictions at the last time point of simulation
+time_index = -1
+
+# use seaborn style
+plt.style.use('seaborn')
+plt.rcParams['image.cmap'] = 'viridis'
-# plot pcolormesh plot of V_e
-im_V_e = ax.pcolormesh(X, Y, V_e.reshape(X.shape),
- shading='auto', cmap='RdBu',
- vmin=-abs(V_e).max() / 2, vmax=abs(V_e).max() / 2,
- zorder=0)
+# create figure and axis
+fig, ax = plt.subplots(1, 1, figsize=(16, 6), dpi=100)
+
+# plot contours of V_e
+lim = float(f'{abs(V_e).max() / 3:.1e}')
+levels = np.linspace(-lim, lim, 25)
+im_V_e = ax.contourf(X, Y, V_e[:, time_index].reshape(X.shape),
+ cmap='RdBu',
+ levels=levels,
+ extend='both')
+
+# V_e colorbar:
cb = colorbar(fig, ax, im_V_e, height=0.45, voffset=0.55)
cb.set_label('$V_e$ (mV)')
-# add outline of each CV
-norm = plt.Normalize(vmin=-66, vmax=-64)
-colors = [plt.cm.viridis(norm(v)) for v in cell_geometry.V_m[:, -1]]
-zips = []
-for i in range(I_m.shape[0]):
- inds = CV_ind == i
- zips.append(create_polygon(x[inds, ].flatten(),
- y[inds, ].flatten(), d[inds, ].flatten()))
-polycol = PolyCollection(zips,
- edgecolors='k',
- facecolors=colors,
- linewidths=0.5,
- zorder=2)
+# add outline of each CV with color coding according to membrane voltage
+vlims = [-66., -64.]
+polycol = get_cv_polycollection(cell_geometry, V_m[time_index, :], vlims=vlims)
im_V_m = ax.add_collection(polycol)
+# V_m colorbar
cb2 = colorbar(fig, ax, im_V_m, height=0.45)
cb2.set_ticks([0, 0.5, 1])
-cb2.set_ticklabels([-66, -65, -64])
+cb2.set_ticklabels([vlims[0], np.mean(vlims), vlims[1]])
cb2.set_label(r'$V_m$ (mV)')
-ax.set_xlim(X.min(), X.max())
-ax.set_ylim(Y.min(), Y.max())
-ax.set_xlabel(r'$x$ ($\mu$m)')
-ax.set_ylabel(r'$y$ ($\mu$m)')
+# draw segment outlines
+ax.add_collection(get_segment_outlines(cell_geometry))
+
+# add marker denoting clamp location
+point = p.at(clamp_location)
+ax.plot(point.x, point.y, 'ko', ms=10, label='stimulus')
+ax.legend()
+
+# axis annotations
+ax.axis(axis)
+ax.set_xlabel(r'$x$ ($\mu$m)', labelpad=0)
+ax.set_ylabel(r'$y$ ($\mu$m)', labelpad=0)
+ax.set_title(f'$V_e$ and $V_m$ at $t$={time[time_index]} ms')
+
+# save file
fig.savefig('single_cell_extracellular_potentials.svg', bbox_inches='tight')
# ## Notes on output:
-# The spatial discretization is here deliberately coarse with only 3 CVs per branch.
+# The spatial discretization is here deliberately coarse with only 3 CVs
+# per branch.
# Hence the branch receiving input about 1/6 of the way from its root
-# (from `decor.place('(location 4 0.16667)', iclamp)`) is treated
-# as 3 separate line sources with homogeneous current density per length unit each, even if
-# the diameter and direction varies with position due to the fact
-# that each CV may be composed of multiple segments.
-#
-# The parameter `nseg = 3` above can be changed above to affect the number
-# of CVs per branch.
+# (from `decor.place('(location 4 0.16667)', iclamp, '"iclamp"')`) is treated
+# as 3 separate line sources with inhomogeneous current density per length
+# unit. This inhomogeneity is due to the fact that the total transmembrane
+# current per CV may distributed across multiple segments with varying surface
+# area. The transmembrane current is assumed to be constant per length unit
+# per segment.