# Distribution of time intervals in Poisson process

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?

• It is not a cumulative distribution function. But if you divide both sides by $\Delta t$ and then take the limit as $\Delta t \to 0_+$ then you get the probability density (the derivative of the CDF) $r \exp(-r\tau)$ Jul 27, 2023 at 10:27
• Thanks for the comment. I considered that possibility, but after dividing by $\Delta t$, shouldn't the left hand side tend to infinity as $\Delta t$, the denominator, approaches $0$? Jul 27, 2023 at 10:32
• Expanding a bit on @Henry's comment: The denominator is also approaching 0, as $\Delta t$ goes to $0$. So you have a ratio that goes to "0/0". Then you can replace both numerator and denominator by their derivative, here with respect to $\Delta t$. This is a consequence of de l'Hôpital's rule. Do you need a complete answer?
– Ute
Jul 27, 2023 at 11:48
• Hello @Ute, thank you for the comment. I can see how l'Hopital applies in theory, but in practice we do not know the derivative of $P[\tau \leq t_{i+1} - t_{i} < \tau + \Delta t]$ with respect to $\Delta t$, do we? Jul 27, 2023 at 11:56
• It is a bit difficult to understand that mixed notation. Does it help if you denote by $D=t_{i+1}-t_i$ the random interspike length and then have $P[\tau \leq D\le \tau+\Delta t]$?
– Ute
Jul 27, 2023 at 12:06

Your difficulties seem to stem from translating the original form of equation (1.31) into the "probability that an interval $$[t+\tau, t+\tau + \Delta t]$$ is an interspike interval". This is due to the authors using the word "interval" synonymously with "length of an interval".

I rewrite the book paragraph using a bit more formalism and hope it becomes easier to understand. I also find it much easier to work with the cdf instead of the pdf.

### Rewrite of book paragraph:

The probability density of length of time intervals between adjacent spikes is called the interspike interval length distribution, and it is a useful statistic for characterizing spiking patterns. Let $$L$$ denote this random length, and $$F$$ be its cumulative density function.

Suppose that a spike occurs at a time $$t_i$$ for some value of $$i$$. The following interspike interval is at least of length $$\tau$$, if there is no new spike in the interval $$(t_i, t_i+\tau)$$. From equation 1.29, with $$n=0$$, the probability of not firing a spike in this period, i.e., the probability that the interspike length exceeds $$\tau$$ is $$P(L > \tau) = \exp (-r\tau).$$ From this we get the cdf of $$L$$, namely $$F(\tau) = P(L\leq \tau) = 1 - \exp(-r\tau).$$ This is the cdf of an exponential distribution with rate parameter $$r$$.

#### New interpretation of book text:

Now look at equation (1.31). Using the symbol $$L$$ for length of interval again, we read $$P(\tau\leq L < \tau+\Delta t) = r \Delta t \exp (-r\tau).$$ This probability can be written as difference of cdf in $$\tau$$ and $$\tau+\Delta t$$, which gives us $$\frac{F(\tau+\Delta t)-F(\tau)}{\Delta t} = r\exp(-r\tau).$$ Now we can take the limit on the left hand side for $$\Delta t \to 0$$ and get the pdf of an exponential distribution.

• There you go - and ask your money back :-D (for the book). Sometimes formal would have been easier than informal...
– Ute
Jul 27, 2023 at 16:39
• Jesus... To be honest I wouldn't have been able to reconstruct the statements in a way that made the logic as clear as you just did. Jul 27, 2023 at 20:43
• Yeah the notation of the book is not excellent :( Jul 27, 2023 at 20:43
• tx - :-) hehe, taught this more than once
– Ute
Jul 27, 2023 at 20:47
• The pedagogical skills do show. Have a great rest of your day Jul 27, 2023 at 21:34