.. currentmodule:: brian

Tutorial 2b: Excitatory and inhibitory currents
***********************************************

In this tutorial, we use multiple connections to solve a real problem, how
to implement two types of synapses with excitatory and inhibitory currents
with different time constants.

The scheme
~~~~~~~~~~

The scheme we implement is the following diffential equations:

    | taum dV/dt = -V + ge - gi
    | taue dge/dt = -ge
    | taui dgi/dt = -gi

An excitatory neuron connects to state ge, and an inhibitory neuron connects
to state gi. When an excitatory spike arrives, ge instantaneously increases,
then decays exponentially. Consequently, V will initially but continuously
rise and then fall. Solving these equations, if V(0)=0, ge(0)=g0 corresponding
to an excitatory spike arriving at time 0, and gi(0)=0 then:

    | gi = 0    
    | ge = g0 exp(-t/taue)
    | V = (exp(-t/taum) - exp(-t/taue)) taue g0 / (taum-taue)

We use a very short time constant for the excitatory currents, a longer one
for the inhibitory currents, and an even longer one for the membrane
potential.

  ::

    from brian import *
    
    taum = 20 * ms
    taue = 1 * ms
    taui = 10 * ms
    Vt = 10 * mV
    Vr = 0 * mV
    
    eqs = Equations('''
          dV/dt  = (-V+ge-gi)/taum : volt
          dge/dt = -ge/taue        : volt
          dgi/dt = -gi/taui        : volt
          ''')
    

Connections
~~~~~~~~~~~

As before, we'll have a group of two neurons under direct control, the first
of which will be excitatory this time, and the second will be inhibitory. To
demonstrate the effect, we'll have two excitatory spikes reasonably close
together, followed by an inhibitory spike later on, and then shortly after
that two excitatory spikes close together.

  ::

    spiketimes = [(0, 1 * ms), (0, 10 * ms),
                  (1, 40 * ms),
                  (0, 50 * ms), (0, 55 * ms)]
    
    G1 = SpikeGeneratorGroup(2, spiketimes)
    G2 = NeuronGroup(N=1, model=eqs, threshold=Vt, reset=Vr)
    
    C1 = Connection(G1, G2, 'ge')
    C2 = Connection(G1, G2, 'gi')

The weights are the same - when we increase ``ge`` the effect on ``V`` is excitatory
and when we increase ``gi`` the effect on ``V`` is inhibitory.

  ::

    C1[0, 0] = 3 * mV
    C2[1, 0] = 3 * mV

We set up monitors and run as normal.

  ::

    Mv = StateMonitor(G2, 'V', record=True)
    Mge = StateMonitor(G2, 'ge', record=True)
    Mgi = StateMonitor(G2, 'gi', record=True)
    
    run(100 * ms)

This time we do something a little bit different when plotting it. We want
a plot with two subplots, the top one will show ``V``, and the bottom one will
show both ``ge`` and ``gi``. We use the ``subplot`` command from pylab which mimics the
same command from Matlab.

  ::

    figure()
    subplot(211)
    plot(Mv.times, Mv[0])
    subplot(212)
    plot(Mge.times, Mge[0])
    plot(Mgi.times, Mgi[0])
    show()

.. image:: images/tutorials/2b.jpg

The top figure shows the voltage trace, and the bottom figure shows ``ge`` in
blue and ``gi`` in green. You can see that although the inhibitory and
excitatory weights are the same, the inhibitory current is much more
powerful. This is because the effect of ``ge`` or ``gi`` on ``V`` is related to the
integral of the differential equation for those variables, and ``gi`` decays
much more slowly than ``ge``. Thus the size of the negative deflection at
40 ms is much bigger than the excitatory ones, and even the double
excitatory spike after the inhibitory one can't cancel it out.

In the next part of this tutorial, we set up our first serious network,
with 4000 neurons, excitatory and inhibitory.

Exercises
~~~~~~~~~

1. Try changing the parameters and spike times to get a feel for how it
   works.
2. Try an equivalent implementation with the equation taum dV/dt = -V+ge+gi
3. Verify that the differential equation has been solved correctly.

Solutions
~~~~~~~~~

Solution for 2:

Simply use the line ``C2[1,0] = -3*mV`` to get the same effect.

Solution for 3:

First, set up the situation we described at the top for which we
already know the solution of the differential equations, by changing
the spike times as follows::

    spiketimes = [(0,0*ms)]

Now we compute what the values ought to be as follows::

    t = Mv.times
    Vpredicted = (exp(-t/taum) - exp(-t/taue))*taue*(3*mV) / (taum-taue)

Now we can compute the difference between the predicted and actual values::

    Vdiff = abs(Vpredicted - Mv[0])

This should be zero::

    print max(Vdiff)

Sure enough, it's as close as you can expect on a computer. When I run this
it gives me the value 1.3 aV, which is 1.3 * 10^-18 volts, i.e. effectively
zero given the finite precision of the calculations involved. 
