# Approximating Bernoulli distributions as Poisson distributions in analysis of neural data

I'm reading Analysis of Neural Data, by Kass et al (2014), where Kass argues that spike trains can be viewed as point processes (in chapter 19). Furthermore, he goes on to convert spike trains (that are events in continuous time) to discrete time by dividing the timeline into bins of equal size.

Now, let $Y_i$ be the numbers of spikes in bin $i$ of size $\Delta t$. Since $\Delta t$ is defined to be small, Kass arrives at the model $Y_i \sim \text{Bernoulli}(p_i)$. And since $p_i$ is also considered to be small, Kass approximates the Bernoulli distributions by Poisson distributions. Kass says that the main reason for this approximation is that this enables him to model the $Y_i$'s in the framework of a GLM since the Poisson regression model is part of the GLM methodology (see page 568).

But wouldn't you say that this last approximation is unnecessary? Why doesn't Kass just use logistic regression? Can anyone comment if Poisson regression is generally preferred in analysing neural data? And if so, why?

• Binomial converges to Poisson so such approximation is often used. Also don't you mean binomial rather then Bernoulli? – Tim Nov 11 '16 at 21:22
• @Tim in this case a sum over the Y_i's will indeed be a random variable, say Z, that has the binomial distribution. Then Z has an approximate Poisson distribution. But my issue with Kass is that he approximates the Y_i's as having Poisson distributions. Am I missing something? – harisf Nov 14 '16 at 21:41
• I can't see any reason for this. Bernoulli distribution is much simpler then Poisson, so I can't see any reason for such approximation (I didn't read the paper). Bernoulli distribution is also a part of GLM methodology (logistic regression) not less then Poisson regression. – Tim Nov 14 '16 at 21:47
• I agree with you. However there seems to be a tradition of approximating, as I've understood @HEITZ's answer. – harisf Nov 15 '16 at 8:05

## 1 Answer

The most likely reason is that spikes are counts, and count data are best modeled by a poisson regression. This doesn't necessarily mean you have to, but in most circumstances it will work out better because you are respecting the form of the response variable.

That said, there are other things you can do to represent your spike train in continuous time. It's pretty common to convolve your spike times with some function yielding a continuous transformation whereby y is now spikes/sec. Gaussian kernels are common, but can be biased. We always we used a 'PSP' or post synaptic potential-like kernel with a sharp rise and decay.

• OK, it makes sense to model spike trains by Poisson regression if they are indeed recorded as counts. However, the data set I'm working with has spikes recorded as event times. I.e. the spike trains are sequences of event times in continuous time already. How would you proceed in analysing such spike trains? Would you say it makes more sense for me to use logistic regression then? – harisf Nov 12 '16 at 12:37
• Well no, the event times must be binned and counted. – HEITZ Nov 12 '16 at 15:26
• Care to explain why they must be binned and counted? – harisf Nov 12 '16 at 19:49
• Most analyses would seek to examine the firing rates between some conditions of interest, no? If so, you have to have some way of saying just how many spikes, in terms of raw counts or a transformation to spikes/time exist within a window of interest. I can't think of any analysis that would use raw spike times, save for maybe an analysis on inter-spike intervals (ISI), but even then you're still creating a derivative measure. – HEITZ Nov 14 '16 at 14:52