Tutorial 1c: Making some activity

In the previous part of the tutorial we found that each neuron was producing only one spike. In this part, we alter the model so that some more spikes will be generated. What we’ll do is alter the resting potential El so that it is above threshold, this will ensure that some spikes are generated. The first few lines remain the same:

from brian import *

tau = 20 * msecond        # membrane time constant
Vt = -50 * mvolt          # spike threshold
Vr = -60 * mvolt          # reset value

But we change the resting potential to -49 mV, just above the spike threshold:

El = -49 * mvolt          # resting potential (same as the reset)

And then continue as before:

G = NeuronGroup(N=40, model='dV/dt = -(V-El)/tau : volt',
              threshold=Vt, reset=Vr)

M = SpikeMonitor(G)

run(1 * second)

print M.nspikes

Running this program gives the output 840. That’s because every neuron starts at the same initial value and proceeds deterministically, so that each neuron fires at exactly the same time, in total 21 times during the 1s of the run.

In the next part, we’ll introduce a random element into the behaviour of the network.

Exercises

1. Try varying the parameters and seeing how the number of spikes generated varies.
2. Solve the differential equation by hand and compute a formula for the number of spikes generated. Compare this with the program output and thereby partially verify it. (Hint: each neuron starts at above the threshold and so fires a spike immediately.)

Solution

Solving the differential equation gives:

V = El + (Vr-El) exp (-t/tau)

Setting V=Vt at time t gives:

t = tau log( (Vr-El) / (Vt-El) )

If the simulator runs for time T, and fires a spike immediately at the beginning of the run it will then generate n spikes, where:

n = [T/t] + 1

If you have m neurons all doing the same thing, you get nm spikes. This calculation with the parameters above gives:

t = 48.0 ms n = 21 nm = 840

As predicted.