I am studying computational neuroscience, particularly the modeling of neuronal spikes. Abstractly, we may think of a spike plainly as some event that either occurs or fails to occur in time. It has been established that the probability that $n$ events (spikes) occur in a trial of time $T$ follows a Poisson distribution with
$$ P_{T}(N = n) = \frac{(rT)^{n}}{n!}\exp(-rT) $$
Here, $r$ is the firing rate of the neuron (or the constant rate at which events occur in time). It was also established early in the book that $r\Delta t$ is the probability that any spike occurs at some time interval $[t, t + \Delta t]$. As a note, the equation above in the book is numbered $1.29$ - I clarify this for it is referenced in the fragment I need help with.
This is all the context required to understand the following fragment of the book, with which I need some help.
I can fully comprehend the paragraph above up to equation $1.31$. What I don't understand is what immediately follows it. Namely, that the PDF of interspike intervals is equation $1.31$ without the factor $\Delta t$, or in other words, that the probability that an interval $[t + \tau, t + \tau + \Delta t]$ is an interspike interval is
$$ r\exp(-r\tau) $$
I did observe that such expression is the derivative with respect to $\Delta t$ of equation $1.31$. However, equation $1.31$ is not a CDF, and hence its derivative is not the PDF we are interested in. What am I missing?
The probability density $f_X$ of a random variable $X$ can be defined via the following limit $$ f_X(x) = \lim_{\Delta \downarrow 0} \frac{P(x < X \leq x+\Delta)}{\Delta}. $$ for a.e. $x$. This results from the Lebesgue differentiation theorem. Now in your case the random variable is $t_{i+1} - t_{i}$ and the $\Delta$ is your $\Delta t$. The density at $\tau$ is then $r \exp(-r\tau)$ since the $\Delta t$ cancels out.