Finally, the update of the lecture.
I will skip lecture VI: I was somehow busy last week, and I could not update the blog.
So, tuesday we started to learn something about networks. In particular, we discussed what can be said about the statistics of the superposition of renewal processes.
Superposition is not a renewal process.
Assume that you have two different processes and fix a spike at time -1 from the first process, the previous one being in a late past. Then, depending on when the last spike of the second process was, the hazard of the superposition will be different, and so it can't be a renewal process. The exception is of course the superposition of Poisson processes for which the hazard is constant.
Superposition of large numbers of renewal processes in equilibrium has exponential distributed intervals
How to see it? Formal method: compute the ISI distribution of the superposition, see the slides or the book by Cox. More intuitevly, you can reason in the following manner: first discretize with width dt the time and assume that a time 0 you had a spike from neuron 1. Now, the probability of having a spike at time 1 will be dt(r_1+r_2+...+r_n) approximately for all subsequent bins: this is because we are superimposing a large number of processes which all are in equilibrium. So, it is a Poisson process in discrete time, and going to the limit dt-->0 we obtain what we are looking for.