Why is the distribution of mean firing rates of a neuron Gaussian? I'm reading "Principles of Neural Information Theory" by James V Stone and in section 3.5 he says that the distribution of firing rates (of a single neuron) is generally assumed to be approximately Gaussian. He proceeds to give a mathematical argument for this.
If we measure the activity during a period of $T$ seconds over intervals $\Delta t$, there are $N=T/ΔtN=T/\Delta t$ possible positions for spikes to occur. The probability of $n$ spikes occurring during this period is
$$p(n)=\frac{N!}{n!(N-n)!}P^nQ^{N-n}$$
Where $P$ is the probability or a spike occurring and $Q$ is the probability of a spike not occuring. If $N$ and $T$ increase while firing rate $r$ stays constant, $p(n)$ approaches the Poisson distribution:
$$p(n)=\frac{(rT)^n e^{-rT}}{n!}$$
If $T$ is held constant then a simple change of variables can be used to obtain a distribution for the firing rate (because $r=n/T$). If $N$ is large and $P$ is small, apparently this distribution is approximated by a Gaussian distribution.
I don't really understand this derivation. It would be nice if someone could provide an in-depth version of the argument with all steps fully fleshed-out. I'm particularly confused about how one goes from p(n) to a Poisson distribution. By looking at the plot, I can see why a Poisson distribution is approximated by a Gaussian distribution, but it would nice to have rigorous justification for this as well.

The asymptotic connections between the binomial, Poisson, and normal distributions are discussed in many textbooks on probability theory.  Here, I'll explain the connection between the two versions of $p(n)$ in your question.
Your first version of $p(n)$, 
$p(n)=\frac{N!}{n!(N-n)!}P^{n}Q^{N-n}$
is simply the binomial probability distribution with parameters $N$ and $P=1-Q$.  
Using the fact that $P+Q=1$, we can rewrite this as 
$p(n)=\frac{N!}{n!(N-n)!}P^{n}(1-P)^{N-n}$.
Let $T$ be our time period and $r$ be the firing rate.  The firing rate is equal to the probability of an individual neuron firing times the number of neurons, divided by the time period $T$.  Thus $r=PN/T$ and $P=rT/N$.  Notice that if $rT$ is kept constant and $N$ increases, $P$ must decrease. Substituting this into our formula for $p(n)$, we get
$p(n)=\frac{N!}{n!(N-n)!}\left( \frac{rT}{N} \right)^{n} \left( 1-\frac{rT}{N} \right)^{N-n}$.
$p(n)=\frac{N!}{n!(N-n)!}\left( \frac{rT}{N} \right)^{n} \left( 1-\frac{rT}{N} \right)^{N} \left( 1-\frac{rT}{N} \right)^{-n}$.
In the limit as $N$ goes to infinity we have (from freshman calculus), 
$\lim_{N \rightarrow \infty} \left( 1- \frac{rT}{N} \right)^{N}=e^{-rT}$.
Also,  
$\lim_{N \rightarrow \infty, P \rightarrow 0} \frac{N!}{(N-n)!} \left( \frac{P}{1-P} \right)^{n}=(rT)^{n}$.
Combining these results gives
$\lim_{N \rightarrow \infty, P \rightarrow 0} p(n)=\frac{(rT)^{n}e^{-rT}}{n!}$.
