Verifying Non-Uniformity of Subvolumes

Here, I sample subvolumes of a predetermined size, count the synapse contents, and then plot that distribution in order to show that the synapses are not uniformly distributed.

If they are uniformly distributed, then the graph of x-center × y-center × count should be a perfectly straight line.



In [1]:

    
%matplotlib inline



In [2]:

    
from matplotlib import pyplot as plt
import numpy as np
import pandas as pd
import seaborn as sns









    



/usr/local/lib/python2.7/site-packages/IPython/html.py:14: ShimWarning: The `IPython.html` package has been deprecated. You should import from `notebook` instead. `IPython.html.widgets` has moved to `ipywidgets`.
  "`IPython.html.widgets` has moved to `ipywidgets`.", ShimWarning)

Import data:



In [3]:

    
import csv

data = open('../data/data.csv', 'r').readlines()
fieldnames = ['x', 'y', 'z', 'unmasked', 'synapses']
reader = csv.reader(data)
reader.next()

rows = [[int(col) for col in row] for row in reader]

sorted_x = sorted(list(set([r[0] for r in rows])))
sorted_y = sorted(list(set([r[1] for r in rows])))
sorted_z = sorted(list(set([r[2] for r in rows])))

vol = np.zeros((len(sorted_x), len(sorted_y), len(sorted_z)))
for r in rows:
    vol[sorted_x.index(r[0]), sorted_y.index(r[1]), sorted_z.index(r[2])] = r[-1]



In [18]:

    
SUBV_SIZE = (10, 10, 5)
SUBV_COUNT = 500
MGN = 15 # margin

Randomly select SUBV_COUNT subvolumes (of size SUBV_SIZE) from the larger volume. Count their contents (sum), and plot x-origin, y-origin, and count (x,y,size).



In [19]:

    
import random
print vol.shape

sample_vol = vol[MGN : -MGN, MGN : -MGN, :]

subvs = []
for i in range(SUBV_COUNT):
    x_origin = random.randint(0, sample_vol.shape[0] - SUBV_SIZE[0])
    y_origin = random.randint(0, sample_vol.shape[1] - SUBV_SIZE[1])
    z_origin = random.randint(0, sample_vol.shape[2] - SUBV_SIZE[2])
    subv = sample_vol[
        x_origin : x_origin + SUBV_SIZE[0],
        y_origin : y_origin + SUBV_SIZE[1],
        z_origin : z_origin + SUBV_SIZE[2]
    ]
    subvs.append((x_origin, y_origin, z_origin, np.sum(subv)))









    



(108, 52, 11)



In [20]:

    
plt.scatter(x=[s[0] for s in subvs], y=[s[1] for s in subvs], c=[s[3]/400 for s in subvs])
plt.xlabel("Dataset x Axis")
plt.ylabel("Dataset y Axis")
plt.suptitle("Synapse count of subvolumes randomly selected across cortex", fontsize="14")









    Out[20]:





<matplotlib.text.Text at 0x1184e3ad0>

From this alone, we can see that the data are nonuniformly distributed. Now let us plot this to characterize the distribution:



In [21]:

    
plt.hist([s[3]/10000 for s in subvs])#, y=[s[1] for s in subvs])
plt.xlabel("Synapse Count in (10x10x10) Supervoxel (x10,000)")
plt.ylabel("Number of supervoxels")
plt.suptitle("Synapse count in randomly selected supervoxels follows nonuniform distribution", fontsize="14")









    Out[21]:





<matplotlib.text.Text at 0x1185387d0>



In [22]:

    
plt.hist2d([s[0] for s in subvs], y=[s[2] for s in subvs])
plt.xlabel("Data x")
plt.ylabel("Data z")
plt.suptitle("Relative synapse sensities are not distributed uniformly over x/z", fontsize=16)









    Out[22]:





<matplotlib.text.Text at 0x118a074d0>

As this 2D histogram above shows, the synapses are not distributed uniformly in XZ-space.

We know already that they are not distributed evenly across the y-axis, as XY and YZ graphs demonstrate below:



In [23]:

    
plt.hist2d([s[0] for s in subvs], y=[s[1] for s in subvs])
plt.xlabel("Data x")
plt.ylabel("Data y")
plt.suptitle("Relative synapse sensities are not distributed uniformly over x/y", fontsize=16)









    Out[23]:





<matplotlib.text.Text at 0x118be3c90>



In [24]:

    
plt.hist2d([s[1] for s in subvs], y=[s[2] for s in subvs])
plt.xlabel("Data y")
plt.ylabel("Data z")
plt.suptitle("Relative synapse sensities are not distributed uniformly over y/z", fontsize=16)









    Out[24]:





<matplotlib.text.Text at 0x118bcccd0>