Note

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# Plot weight matrices example¶

This example demonstrates how to extract the connection strength for all the synapses among two populations of neurons and gather these values in weight matrices for further analysis and visualization.

All connection types between these populations are considered, i.e., four weight matrices are created and plotted.

First, we import all necessary modules to extract, handle and plot the connectivity matrices

```
import numpy as np
import matplotlib.pyplot as plt
import nest
import matplotlib.gridspec as gridspec
from mpl_toolkits.axes_grid1 import make_axes_locatable
```

We now specify a function to extract and plot weight matrices for all connections among E_neurons and I_neurons.

We initialize all the matrices, whose dimensionality is determined by the number of elements in each population. Since in this example, we have 2 populations (E/I), \(2^2\) possible synaptic connections exist (EE, EI, IE, II).

Note the use of “post-pre” notation when referring to synaptic connections. As a matter of convention in computational neuroscience, we refer to the connection from inhibitory to excitatory neurons (I->E) as EI (post-pre) and connections from excitatory to inhibitory neurons (E->I) as IE (post-pre).

```
def plot_weight_matrices(E_neurons, I_neurons):
W_EE = np.zeros([len(E_neurons), len(E_neurons)])
W_EI = np.zeros([len(I_neurons), len(E_neurons)])
W_IE = np.zeros([len(E_neurons), len(I_neurons)])
W_II = np.zeros([len(I_neurons), len(I_neurons)])
a_EE = nest.GetConnections(E_neurons, E_neurons)
# Using `get`, we can extract the value of the connection weight,
# for all the connections between these populations
c_EE = a_EE.weight
# Repeat the two previous steps for all other connection types
a_EI = nest.GetConnections(I_neurons, E_neurons)
c_EI = a_EI.weight
a_IE = nest.GetConnections(E_neurons, I_neurons)
c_IE = a_IE.weight
a_II = nest.GetConnections(I_neurons, I_neurons)
c_II = a_II.weight
# We now iterate through the range of all connections of each type.
# To populate the corresponding weight matrix, we begin by identifying
# the source-node_id (by using .source) and the target-node_id.
# For each node_id, we subtract the minimum node_id within the corresponding
# population, to assure the matrix indices range from 0 to the size of
# the population.
# After determining the matrix indices [i, j], for each connection
# object, the corresponding weight is added to the entry W[i,j].
# The procedure is then repeated for all the different connection types.
a_EE_src = a_EE.source
a_EE_trg = a_EE.target
a_EI_src = a_EI.source
a_EI_trg = a_EI.target
a_IE_src = a_IE.source
a_IE_trg = a_IE.target
a_II_src = a_II.source
a_II_trg = a_II.target
for idx in range(len(a_EE)):
W_EE[a_EE_src[idx] - min(E_neurons),
a_EE_trg[idx] - min(E_neurons)] += c_EE[idx]
for idx in range(len(a_EI)):
W_EI[a_EI_src[idx] - min(I_neurons),
a_EI_trg[idx] - min(E_neurons)] += c_EI[idx]
for idx in range(len(a_IE)):
W_IE[a_IE_src[idx] - min(E_neurons),
a_IE_trg[idx] - min(I_neurons)] += c_IE[idx]
for idx in range(len(a_II)):
W_II[a_II_src[idx] - min(I_neurons),
a_II_trg[idx] - min(I_neurons)] += c_II[idx]
fig = plt.figure()
fig.subtitle('Weight matrices', fontsize=14)
gs = gridspec.GridSpec(4, 4)
ax1 = plt.subplot(gs[:-1, :-1])
ax2 = plt.subplot(gs[:-1, -1])
ax3 = plt.subplot(gs[-1, :-1])
ax4 = plt.subplot(gs[-1, -1])
plt1 = ax1.imshow(W_EE, cmap='jet')
divider = make_axes_locatable(ax1)
cax = divider.append_axes("right", "5%", pad="3%")
plt.colorbar(plt1, cax=cax)
ax1.set_title('W_{EE}')
plt.tight_layout()
plt2 = ax2.imshow(W_IE)
plt2.set_cmap('jet')
divider = make_axes_locatable(ax2)
cax = divider.append_axes("right", "5%", pad="3%")
plt.colorbar(plt2, cax=cax)
ax2.set_title('W_{EI}')
plt.tight_layout()
plt3 = ax3.imshow(W_EI)
plt3.set_cmap('jet')
divider = make_axes_locatable(ax3)
cax = divider.append_axes("right", "5%", pad="3%")
plt.colorbar(plt3, cax=cax)
ax3.set_title('W_{IE}')
plt.tight_layout()
plt4 = ax4.imshow(W_II)
plt4.set_cmap('jet')
divider = make_axes_locatable(ax4)
cax = divider.append_axes("right", "5%", pad="3%")
plt.colorbar(plt4, cax=cax)
ax4.set_title('W_{II}')
plt.tight_layout()
```

The script iterates through the list of all connections of each type. To populate the corresponding weight matrix, we identify the source-node_id (first element of each connection object, n[0]) and the target-node_id (second element of each connection object, n[1]). For each node_id, we subtract the minimum node_id within the corresponding population, to assure the matrix indices range from 0 to the size of the population.

After determining the matrix indices [i, j], for each connection object, the corresponding weight is added to the entry W[i,j]. The procedure is then repeated for all the different connection types.

We then plot the figure, specifying the properties we want. For example, we
can display all the weight matrices in a single figure, which requires us to
use `GridSpec`

to specify the spatial arrangement of the axes.
A subplot is subsequently created for each connection type. Using `imshow`

,
we can visualize the weight matrix in the corresponding axis. We can also
specify the colormap for this image.
Using the `axis_divider`

module from `mpl_toolkits`

, we can allocate a small
extra space on the right of the current axis, which we reserve for a
colorbar.
We can set the title of each axis and adjust the axis subplot parameters.
Finally, the last three steps are repeated for each synapse type.

**Total running time of the script:** ( 0 minutes 0.000 seconds)