I want to get familiar with the brian2 neuron and neuron populations simulator; hence, this notebook.

Part I: Neurons

Imports and Setup

In [1]:
%matplotlib inline

from brian2 import *

brian2 has a system for using quantities with physical dimensions. All the basic SI units can be used along with their standard prefixes, and some standard abbreviations:

In [2]:
20 * volt
Out[2]:
$20.0\,\mathrm{V}$
In [3]:
1000 * amp
Out[3]:
$1.0\,\mathrm{k}\,\mathrm{A}$
In [4]:
100 * namp
Out[4]:
$100.0\,\mathrm{n}\,\mathrm{A}$
In [5]:
10 * nA * 5 * Mohm
Out[5]:
$50.0\,\mathrm{m}\,\mathrm{V}$

This should produce a DimensionMismatchError:

In [6]:
5 * amp + 10 * volt
---------------------------------------------------------------------------
DimensionMismatchError                    Traceback (most recent call last)
<ipython-input-6-efddc2e2a7d8> in <module>()
      1 # this should produce a DimensionMismatchError:
----> 2 5 * amp + 10 * volt

/home/dan/anaconda3/envs/py27/lib/python2.7/site-packages/brian2/units/fundamentalunits.pyc in __add__(self, other)
   1412         return self._binary_operation(other, operator.add,
   1413                                       fail_for_mismatch=True,
-> 1414                                       operator_str='+')
   1415 
   1416     def __radd__(self, other):

/home/dan/anaconda3/envs/py27/lib/python2.7/site-packages/brian2/units/fundamentalunits.pyc in _binary_operation(self, other, operation, dim_operation, fail_for_mismatch, operator_str, inplace)
   1352                 _, other_dim = fail_for_dimension_mismatch(self, other, message,
   1353                                                            value1=self,
-> 1354                                                            value2=other)
   1355 
   1356         if other_dim is None:

/home/dan/anaconda3/envs/py27/lib/python2.7/site-packages/brian2/units/fundamentalunits.pyc in fail_for_dimension_mismatch(obj1, obj2, error_message, **error_quantities)
    183             raise DimensionMismatchError(error_message, dim1)
    184         else:
--> 185             raise DimensionMismatchError(error_message, dim1, dim2)
    186     else:
    187         return dim1, dim2

DimensionMismatchError: Cannot calculate 5. A + 10. V, units do not match (units are amp and volt).

A simple model

Let's start by defining a simple neuron model. In brian2, all models are defined by systems of differential equations. Here's a simple example:

In [7]:
tau = 10*ms
eqs = '''
dv/dt = (1-v)/tau : 1
'''

Equations are given by a multi-line string with one equation per string. The equations are formatted with standard mathematical notation, but with one exception: at the end of each line, we write : unit where unit is the SI unit of the variable in the differential equation.

Let's use this definition to create a neuron:

In [8]:
G = NeuronGroup(1, eqs, method='linear')

In brian2, we can only create groups of neurons using the class NeuronGroup. The first two arguments when you create one of these objects are the number of neurons and their defining differential equations.

The command run(100*ms) runs the simulation for 100ms. We can see that it has worked by printing out the value of v before and after the simulation.

In [9]:
print('Before v = %s' % G.v[0])
run(100*ms)
print('After v = %s' % G.v[0])
Before v = 0.0
After v = 0.99995460007

By default, all variables start with value 0. Since the differential equation is dv/dt = (1-v)/tau, we would expect after a while that v would tend towards 1, which is what we've observed. Specifically, we should expect to see that v has the value 1-exp(-t/tau). Let's check this:

In [10]:
print('Expected value of v = %s' % (1 - exp(-100 * ms / tau)))
Expected value of v = 0.99995460007

The simulation gives the expected value!

Now, let's look at a plot of how the value v evolves over time:

In [11]:
# so this code doesn't depend on previously defined variables
start_scope()

G = NeuronGroup(1, eqs, method='linear')
M = StateMonitor(G, 'v', record=0)

run(30*ms)

plot(M.t / ms, M.v[0])
xlabel('Time (ms)');
ylabel('v');
title('Voltage over Time');

We ran the simulation only for 30ms so we can see the behavior better. It looks like its behaving as expected, but let's check it analytically by plotting the expected behavior as well.

In [12]:
start_scope()

G = NeuronGroup(1, eqs, method='linear')
M = StateMonitor(G, 'v', record=0)

run(30*ms)

plot(M.t/ms, M.v[0], '-b', lw=2, label='Brian')
plot(M.t/ms, 1-exp(-M.t/tau), '--r', lw=2, label='Analytic')
xlabel('Time (ms)');
ylabel('v');
legend(loc='best');
title('Voltage over Time');

Clearly, the blue line (brian2 simulation) and the red line (analytic solution) coincide.

Now, we modify the equations and its parameters and see what happens in the plot of the simulation:

In [13]:
start_scope()

tau = 10*ms
eqs = '''
dv/dt = (sin(2*pi*100*Hz*t)-v)/tau : 1
'''

# Change to Euler's method since exact integration doesn't work for this differential equation
G = NeuronGroup(1, eqs, method='euler')
M = StateMonitor(G, 'v', record=0)

# Initial value for v
G.v = 5

run(60*ms)

plot(M.t/ms, M.v[0])
xlabel('Time (ms)');
ylabel('v');
title('Voltage over Time');

Adding Spikes

So far we haven't done anything neuronal, just played around with differential equations. Let's start adding spiking behavior.

In [14]:
start_scope()

tau = 10*ms
eqs = '''
dv/dt = (1-v)/tau : 1
'''

# creating a single neuron which "spikes" at v = 0.8 and resets to v = 0
G = NeuronGroup(1, eqs, threshold='v>0.8', reset='v=0', method='linear')
M = StateMonitor(G, 'v', record=0)

run(50*ms)
plot(M.t/ms, M.v[0])
xlabel('Time (ms)')
ylabel('v')
title('Spiking Neurons');

We've added two keywords to the NeuronGroup declaration: threshold='v>0.8' and reset='v=0'. This means that whenever v>0.8 we "fire" a spike and immediately reset v=0 afterwards. We can put any expression or series of statements as these arguments.

At the beginning of the simulation, the behavior is the same as before until v crosses the threshold of v>0.8, at which point v resets to 0. You can't see this in the figure, but internally, brian2 has registered this as a spike. Let's take a look at this.

In [15]:
start_scope()

G = NeuronGroup(1, eqs, threshold='v>0.8', reset='v=0', method='linear')

# creating a 'SpikeMonitor' object to record spike activity
spikemon = SpikeMonitor(G)

run(50*ms)

print('Spike times: %s' % spikemon.t[:])
creating /tmp/scipy-dan-JIpGqd/python27_intermediate/compiler_dc7ecac41cfd71b2db4283a38d127b00
Spike times: [ 16.   32.1  48.2] ms

The SpikeMonitor object takes the group whose spikes we'd like to record as its argument, and stores the spike tiems in the variable t. Let's plot the spikes on top of the other figure to see that it's getting the times right.

In [16]:
start_scope()

G = NeuronGroup(1, eqs, threshold='v>0.8', reset='v=0', method='linear')

# creating both 'StateMonitor' and 'SpikeMonitor' objects
statemon = StateMonitor(G, 'v', record=0)
spikemon = SpikeMonitor(G)

run(50*ms)

plot(statemon.t/ms, statemon.v[0])
for t in spikemon.t:
    axvline(t/ms, ls='--', c='r', lw=3)
xlabel('Time (ms)')
ylabel('v')
title('Voltage Behavior and Spikes');

Let's try a few more settings for the threshold and reset arguments.

In [17]:
start_scope()

G = NeuronGroup(1, eqs, threshold='v>0.6', reset='v=0.1', method='linear')

# creating both 'StateMonitor' and 'SpikeMonitor' objects
statemon = StateMonitor(G, 'v', record=0)
spikemon = SpikeMonitor(G)

run(50*ms)

plot(statemon.t/ms, statemon.v[0])
for t in spikemon.t:
    axvline(t/ms, ls='--', c='r', lw=3)
xlabel('Time (ms)')
ylabel('v')
title('Voltage Behavior and Spikes');
In [18]:
start_scope()

G = NeuronGroup(1, eqs, threshold='v>0.6', reset='v=-0.3', method='linear')

# creating both 'StateMonitor' and 'SpikeMonitor' objects
statemon = StateMonitor(G, 'v', record=0)
spikemon = SpikeMonitor(G)

run(50*ms)

plot(statemon.t/ms, statemon.v[0])
for t in spikemon.t:
    axvline(t/ms, ls='--', c='r', lw=3)
xlabel('Time (ms)')
ylabel('v')
title('Voltage Behavior and Spikes');

Refractoriness

A common feature of neuron models is refractoriness; i.e., after a neuron fires, it become refractory, or inactive, for a certain duration and cannot fire another spike until this period is over. Here's how we do that in brian2:

In [19]:
start_scope()

# setting up equations and time scale
tau = 10*ms
eqs = '''
dv/dt = (1-v)/tau : 1 (unless refractory)
'''

# simulating one neuron with the same 'threshold' and 'reset' args, but with a 5ms refractory period
G = NeuronGroup(1, eqs, threshold='v>0.8', reset='v=0', refractory=5*ms, method='linear')

# creating state and spike monitor objects
statemon = StateMonitor(G, 'v', record=0)
spikemon = SpikeMonitor(G)

run(50*ms)

plot(statemon.t/ms, statemon.v[0])
for t in spikemon.t:
    axvline(t/ms, ls='--', c='r', lw=3)
xlabel('Time (ms)')
ylabel('v')
title('Single Neuron Spikes with Refractoriness');

From this figure, after the first spike, we see that v stays at 0 for around 5ms before it resumes its normal behavior. To do this, we've done two things: added the keyword refractory=5*ms to the NeuronGroup declaration, and added the (unless refractory) to the end of the definition of v in the differential equations. The first means only that v cannot spike in the refractory period, and the second switches off the evolution of the differential equation when the neuron is refractory.

Here's what would happen if we didn't include the (unless refractory). Note that we've also decreased the value of tau and increased the length of the refractory period to make the behavior clear.

In [20]:
start_scope()

# setting up time scale and differential equations
tau = 5*ms
eqs = '''
dv/dt = (1-v)/tau : 1
'''

# creating a single neuron with same 'threshold' and 'reset' args, and 15ms refractory period
G = NeuronGroup(1, eqs, threshold='v>0.8', reset='v=0', refractory=15*ms, method='linear')

# setting up state and spike monitor objects
statemon = StateMonitor(G, 'v', record=0)
spikemon = SpikeMonitor(G)

run(50*ms)

plot(statemon.t/ms, statemon.v[0])
for t in spikemon.t:
    axvline(t/ms, ls='--', c='r', lw=3)
axhline(0.8, ls=':', c='g', lw=3)
xlabel('Time (ms)')
ylabel('v')
title('Single Neurons Spikes With Odd Refractory Behavior');
print("Spike times: %s" % spikemon.t[:])
Spike times: [  8.   23.1  38.2] ms

What's going on here? The behavior for the first spike is the same: v rises to 0.8 and then fires a spike at time 8ms before immediately resetting back to 0. Since the refractory period is no 15ms, this means the neuron won't be able to spike again until time 8ms + 15ms = 23ms. Immediately after the spike, the value of v again begins to rise since we didn't specify (unless refractory) in the definition of dv/dt. Once it reaches the threshold value of 0.8, however, it does not fire a spike even though it exceeds the v>0.8 threshold. This is because the neuron is still refractory until time 23ms, at which point it fires a spike.

See the full brian2 documentation for more options and use cases with spikes.

Multiple Neurons

So far we've only being working with a single neuron. Let's do something interesting with multiple neurons!

In [21]:
start_scope()

# using 100 neurons, 10ms time scale, and slightly different 
# differential equation for simulation
N = 100
tau = 10*ms
eqs = '''
dv/dt = (2-v)/tau : 1
'''

# creating the neuron group w/ voltage threshold at 1 and resetting to 0
G = NeuronGroup(N, eqs, threshold='v>1', reset='v=0', method='linear')

# instantiate each neuron with a random voltage
G.v = 'rand()'

# create spike monitor object
spikemon = SpikeMonitor(G)

run(50*ms)

plot(spikemon.t/ms, spikemon.i, '.k')
xlabel('Time (ms)')
ylabel('Neuron index')
title('N = 100 Neuron Population Spikes');

We have a new variable N which determines the number of neurons, we've added the statement G.v = rand() before the run (which inititializes each neurons with a random value uniformly in $\[0, 1\]$, so each neuron does something a bit different), and we've plotted the spikes of each neuron by index over time.

As well as the variable SpikeMonitor.t (which gives the times of all the spikes), we've also used the variable SpikeMonitor.i, which gives the corresponding neuron index for each spike. We plotted a single black dot with time on the x-axis and neuron index on the y-axis. This is the standard "raster plot" used in neuroscience.

Parameters

To make these multiple neurons do something more interesting, let's introduce per-neuron parameters that don't have a differential equation attached to them.

In [22]:
start_scope()

# number of neurons, time scale, initial voltage maximum, and duration of simulation
N = 100
tau = 10*ms
v0_max = 3.
duration = 1000*ms

# differential equations for neurons
eqs = '''
dv/dt = (v0-v)/tau : 1 (unless refractory)
v0 : 1
'''

# creating neuron group of size N with corresponding equations,
# threshold and reset at 1, 0, and 5ms refractory
G = NeuronGroup(N, eqs, threshold='v>1', reset='v=0', refractory=5*ms, method='linear')
M = SpikeMonitor(G)

# initial voltage based on neuron index
G.v0 = 'i*v0_max/(N-1)'

run(duration)

figure(figsize=(12,4))
subplot(121)
plot(M.t/ms, M.i, '.k')
xlabel('Time (ms)')
ylabel('Neuron index')
subplot(122)
plot(G.v0, M.count/duration)
xlabel('v0')
ylabel('Firing rate (sp/s)');

The line v0 : 1 declares a new per-neuron parameter v0 with units 1 (dimensionless). The line G.v0 = 'i*v0_max/(N-1)' initializes the value of v0 for each neuron, varying from 0 up to v0_max. The symbol i, when it appears in equations like this, refers to the index of a neuron.

In this example, we're driving the neuron towards the value v0 exponentially quickly, but we fire spikes when v crosses v>1 and resets to 0. The effect is that the rate at which each neuron fires spikes will be related to the value of v0. For v0<1 it will never fire a spike, and as v0 gets larger it will fire spikes at a higher rate. The right hand plot show the firing rate as a function of the value of v0. This is called the "I-f curve" of this neuron model.

Note that in the plot we've used the count variable of the SpikeMonitor. This is an array of the number of spikes each neuron in the group fired. Dividing this by the duration of the run gives the firing rate.

Stochastic Neurons

Often when making models of neurons, we include a random element to model the effect of various forms of neural noise. In brian2, we can do this by using the symbol xi in differential equations. Strictly speaking, this symbol is a stochastic differential, but you can think of it as a Gaussian random variable with mean 0 and standard deviation 1. We do have to take into account the way stochastic differentials scale with time, which is why we multiply it by tau ** -0.5 in the equations below.

In [23]:
start_scope()

# number of neurons, time scale, max starting voltage, duration of simulation, and 'sigma' (?)
N = 100
tau = 10*ms
v0_max = 3.
duration = 1000*ms
sigma = 0.2

# differential equations (now with stochastic differential (SD) for modeling neural noise)
eqs = '''
dv/dt = (v0-v)/tau+sigma*xi*tau**-0.5 : 1 (unless refractory)
v0 : 1
'''

# creating group of neurons to simulate (solving with Euler's method (possible due to SD))
G = NeuronGroup(N, eqs, threshold='v>1', reset='v=0', refractory=5*ms, method='euler')
# creating spike monitor object on G
M = SpikeMonitor(G)

# specifying firing rate for each neuron i in [1, ..., N]
G.v0 = 'i*v0_max/(N-1)'

run(duration)

figure(figsize=(12,4))
subplot(121)
plot(M.t/ms, M.i, '.k')
xlabel('Time (ms)')
ylabel('Neuron index')
subplot(122)
plot(G.v0, M.count/duration)
xlabel('v0')
ylabel('Firing rate (sp/s)');

This simulation ends up given the same figure as in the last simulation, but with some noise added. Note how the firing rate vs v0 curve has changed shape: instead of a sharp jump from firing at rate 0 to firing at a positive rate, it now increases in a sigmoidal (S-shaped) fashion. This is because no matter how small the driving force, the randomness may cause a neuron to fire a spike.

End of Part I

This is the end of the "Neurons" part of the tutorial. The cell below has an additional example, which we inspect in order to find out what it accomplishes and why.

In [24]:
start_scope()

# number of neurons, time scale, 'vr' (?), 'vt0' (?), 'delta_vt0' (?), 'tau_t' (?),
# 'sigma' (?), 'v_drive' (?), duration of simulation
N = 1000
tau = 10*ms
vr = -70*mV
vt0 = -50*mV
delta_vt0 = 5*mV
tau_t = 100*ms
sigma = 0.5*(vt0-vr)
v_drive = 2*(vt0-vr)
duration = 100*ms

# differential equations (dv/dt and dvt/dt (?))
eqs = '''
dv/dt = (v_drive+vr-v)/tau + sigma*xi*tau**-0.5 : volt
dvt/dt = (vt0-vt)/tau_t : volt
'''

# reset equations
reset = '''
v = vr
vt += delta_vt0
'''

# creating group of 1000 neurons and spike monitor object
G = NeuronGroup(N, eqs, threshold='v>vt', reset=reset, refractory=5*ms, method='euler')
spikemon = SpikeMonitor(G)

# setting initial variable states
G.v = 'rand()*(vt0-vr)+vr'
G.vt = vt0

run(duration)

_ = hist(spikemon.t/ms, 100, histtype='stepfilled', facecolor='k', weights=ones(len(spikemon))/(N*defaultclock.dt))
xlabel('Time (ms)')
ylabel('Instantaneous firing rate (sp/s)');

Part II: Synapses

The Simplest Synapse

Once we have the neurons, the next step is to connect them up with synapses. We'll start by using the simplest possible type of synapse, which causes an instantaneous change in the postsynaptic neuron membrane potential after the spike.

In [29]:
start_scope()

# differential equations
eqs = '''
dv/dt = (I-v)/tau : 1
I : 1
tau : second
'''

# setting up neuron group, current, and time scales
G = NeuronGroup(2, eqs, threshold='v>1', reset='v=0', method='linear')
G.I = [2, 0]
G.tau = [10, 100] * ms

# set up and connect a single synapse from first to second neuron
S = Synapses(G, G, on_pre='v_post += 0.2')
S.connect(i=0, j=1)

# set up state monitor object
M = StateMonitor(G, 'v', record=True)

run(100*ms)

# plotting results of simulation
plot(M.t/ms, M.v[0], '-b', label='Neuron 0')
plot(M.t/ms, M.v[1], '-g', lw=2, label='Neuron 1')
xlabel('Time (ms)')
ylabel('v')
legend(loc='best')
title('Single Excitatory Synapse');

There are a few things going on here. Let's recap what's going on with the NeuronGroup first. We've created two neurons, each of which has the same differential equation but different values for parameters I and tau. Neuron 0 has I = 2 and tau = 10 * ms, which means that it's driven to repeatedly spike at a fairly high rate. Neuron 1 has I = 0 and tau = 100 * ms, which means that on its own (without the synapse) it won't spike at all (the driving current I is 0). We can see this by commenting out the lines that define the synapse.

In [30]:
start_scope()

# differential equations
eqs = '''
dv/dt = (I-v)/tau : 1
I : 1
tau : second
'''

# setting up neuron group, current, and time scales
G = NeuronGroup(2, eqs, threshold='v>1', reset='v=0', method='linear')
G.I = [2, 0]
G.tau = [10, 100] * ms

# set up and connect a single synapse from first to second neuron
# S = Synapses(G, G, on_pre='v_post += 0.2')
# S.connect(i=0, j=1)

# set up state monitor object
M = StateMonitor(G, 'v', record=True)

run(100*ms)

# plotting results of simulation
plot(M.t/ms, M.v[0], '-b', label='Neuron 0')
plot(M.t/ms, M.v[1], '-g', lw=2, label='Neuron 1')
xlabel('Time (ms)')
ylabel('v')
legend(loc='best')
title('Single Excitatory Synapse');

Next, we define the synapses: Synapses(source, targets, ...) means that we are defining a synaptic model that goes from source to target. In this case, the source and target are both the same, the group G. The syntax on_pre='v_post += 0.2' means that when a spike occurs in the presynaptic neuron (hence on_pre), it causes an instantaneous change to happen v_post += 0.2. The _post means that the value of v referred to is the post-synaptic value, and it is to be increased by 0.2. In short, this model says that whenever two neurons in G are connected by a synapse, when the source neuron fires a spike, the target neuron will have its value of v increased by 0.2.

At this point we have only defined the synapse model, but we haven't actually created any synapses. The next line (S.connect(i=0, j=1)) creates a synapse from neuron 0 to neuron 1.

Adding a Weight

In the previous section, we hardcoded the weight of the synapse to be the value 0.2, but we often want this to be different for different synapses. We do this by introducing synapse equations.

In [47]:
start_scope()

# define differential equations
eqs = '''
dv/dt = (I-v)/tau : 1
I : 1
tau : second
'''

# defining neuron group, currents, and time scales
G = NeuronGroup(3, eqs, threshold='v>1', reset='v=0', method='linear')
G.I = [2, 0, 0]
G.tau = [10, 100, 100] * ms

# defining synapses and their connections and weights
S = Synapses(G, G, 'w : 1', on_pre='v_post += w')
S.connect(i=0, j=[1, 2])
S.w = 'j * 0.2'

# state monitor object
M = StateMonitor(G, 'v', record=True)

run(50*ms)

plot(M.t/ms, M.v[0], '-b', label='Neuron 0')
plot(M.t/ms, M.v[1], '-g', lw=2, label='Neuron 1')
plot(M.t/ms, M.v[2], '-r', lw=2, label='Neuron 2')
xlabel('Time (ms)')
ylabel('v')
legend(loc='best')
title('Two Synapses with Varying Weights');

This example behaves similarly to the previous example, but now there's a synaptic weight variable w. The string w : 1 is an equation string, exactly the same as for neurons, which defines a single dimensionless parameter w. We changed the behavior on a spike to on_pre='v_post += w' so that each synapse can behave differently depending on the value of w. To see this, we've made a third neuron which behaves the same as the second neuron, and connected neuron 0 to both neurons 1 and 2. We've set the weights via S.w = 'j * 0.2'. When i and j occur in the context of synapses, i refers to the presynaptic (source) neuron index, and j to the postsynaptic (target) neuron index. So, this will give a synaptic connection from 0 to 1 with weight 0.2 = 0.2 * 1, and from 0 to 2 with weight 0.4 = 0.2 * 2.

Introducing a Delay

So far the synapses have transmitted action potentials instantaneously, but we can also make them act with a certain delay.

In [49]:
start_scope()

# setting up differential equations
eqs = '''
dv/dt = (I-v)/tau : 1
I : 1
tau : second
'''

# setting up neuron group, driving currents, and time scales
G = NeuronGroup(3, eqs, threshold='v>1', reset='v=0', method='linear')
G.I = [2, 0, 0]
G.tau = [10, 100, 100] * ms

# setting up synapses, connections, weights, and delay
S = Synapses(G, G, 'w : 1', on_pre='v_post += w')
S.connect(i=0, j=[1, 2])
S.w = 'j * 0.2'
S.delay = 'j * 2 * ms'

# state monitor object
M = StateMonitor(G, 'v', record=True)

run(50*ms)

plot(M.t/ms, M.v[0], '-b', label='Neuron 0')
plot(M.t/ms, M.v[1], '-g', lw=2, label='Neuron 1')
plot(M.t/ms, M.v[2], '-r', lw=2, label='Neuron 1')
xlabel('Time (ms)')
ylabel('v')
legend(loc='best')
title('Two Synapses with Varying Weights and Delays');

Adding delays is as simple as adding the line S.delay = 'j * 2 * ms', which causes the synapse from 0 to 1 to have a propagating delay of 1 2ms = 2ms, and the synapse from 0 to 2 to have a propagating delay of 2 2ms = 4ms.

More Complex Connectivity

So far, we've specified the connectivity explicitly, but for larger networks this isn't usually possible. For that, we shall want to specify some condition on the connectivity.

In [50]:
start_scope()

N = 10 # number of neurons
G = NeuronGroup(N, 'v : 1') # neuron group object
S = Synapses(G, G) # synapses from G to G
S.connect(condition='i != j', p=0.2) # connectivity pattern -> all to all others each with probability 0.2

We've created a dummy neuron group of N neurons and a dummy synapses model which doesn't actually do anything, just to demonstrate the connectivity. The line S.connect(condition='i != j', p=0.2) will connect all pairs of neurons i to j with probability 0.2 as long as the condition i != j holds. How can we visualize this connectivity? Here is a function that will let us do so:

In [55]:
def visualize_connectivity(S):
    Ns = len(S.source) # number of source neurons
    Nt = len(S.target) # number of target neurons
    figure(figsize=(10, 4))
    subplot(121)
    plot(zeros(Ns), arange(Ns), 'ok', ms=10) # draw source neurons
    plot(ones(Nt), arange(Nt), 'ok', ms=10) # draw target neurons
    for i, j in zip(S.i, S.j):
        plot([0, 1], [i, j], '-k') # draw lines between them
    xticks([0, 1], ['Source', 'Target'])
    ylabel('Neuron index')
    xlim(-0.1, 1.1)
    ylim(-1, max(Ns, Nt))
    title('Connectivity: Source to Target')
    subplot(122)
    plot(S.i, S.j, 'ok')
    xlim(-1, Ns)
    ylim(-1, Nt)
    xlabel('Source neuron index')
    ylabel('Target neuron index')
    title('Connectivity Matrix')
    
visualize_connectivity(S)