# Why Poisson Process for neural firing?

I am new to the world of neuroscience and I would like to understand how come every paper in neuroscience that I read uses a Poisson process to model neural firing. Do I just have a biased sample of the papers I chose to read, or is it a biological fact that neural firing must follow a Poisson process? Why can't it follow, say, binomial, since it either spikes or it doesn't in a given time period. Also, where can I read more about how such a biological phenomena was discovered?

A small correction: I do appreciate the explanation on the difference between the two distributions but I am a statistician who is trying to understand my neuroscience colleagues and communication is key in statistics. I am just trying to comprehend why everyone in the field is using Poisson as opposed to using Binomial even when the bin size you decided on is sufficiently small. In other words, why would you not use Binomial, the bin size is so small that your data is only 0's and 1's (i.e. the spike happened or did not happen within this bin). Is it because Poisson counts discrete occurrences in a continuous domain? What discipline-specific papers would you recommend to read that justify/discuss/explain the continuous domain on neural spiking?

• At the limit of small bin size, there isn't much difference. For large bin size, Poisson works better since you don't need to know the max # of events. Also there are plenty of non-Poisson point process models out there in neuroscience. I tend to use autoregressive point process models. – Memming Mar 27 '17 at 12:51

You can read about the differences between a binomial and poisson distribution at this other question on Math.SE.

First, since spikes are discrete events, it makes most sense to use a probability distribution that is discrete. A Poisson process is a process where an event occurs randomly with no "memory" for how long it has been since the last event. A Poisson distribution is the number of events if you integrate draws above some threshold from an infinitesimally small uniform distribution. Therefore, the Poisson distribution is the distribution you choose if occurrences of an event are only dictated by an underlying rate parameter - this turns out to be a reasonable, if ultimately false, assumption for neurons.

There is definitely no biological fact that neurons must follow a Poisson distribution - in fact, no neuron can be truly Poisson because, if nothing else, the refractory period makes it impossible. Rather, Poisson behavior is sort of the simplest behavior you would expect from neurons in a system where they are firing occasionally due to random noise - sort of a "null hypothesis." For many modeling approaches, this is close enough, and it allows the modeler to avoid making more nuanced assumptions. It turns out to be a decent assumption when you look at individual cells, but if you look at populations the activity deviates a lot from Poisson.

Note that the Poisson distribution is often only valid for spontaneous data; it is possible to use a nonstationary Poisson distribution to model responses to some external stimulus, but the results are likely to deviate more from experimentally derived data.

• Thank you for your response. I am a statistician and discreteness and independence are also the features of Binomial distribution, so even in those cases when bin size is small enough that you data is 1's and 0's why would you not use Binomial? Is it because Poisson counts discrete occurrences in a continuous domain? – Jen Mar 27 '17 at 12:16
• and wouldn't it make more sense to use Binomial if there is a refractory period between spikes? Sorry, I am just really lost on the whole process of neural spiking even after 1 MOOC course and a ton of Youtube videos (I made sure those are from reputable sources at least) – Jen Mar 27 '17 at 12:26
• @Jen A poisson is essentially equivalent to integrating a binomial with infinitely brief windows (and a correspondingly infinitely small probability). Neither can really account for a refractory period. For spike rates in most typical neurons, the rates are low enough that you rarely generate spikes within a refractory period, but an alternative is to generate interspike intervals (ISI) from an exponential distribution (this is exactly the same as drawing Poisson counts), and discard very short intervals (this part deviates from Poisson slightly). – Bryan Krause Mar 27 '17 at 14:43
• @Jen Poisson and binomial become nearly identical for small enough bins but the larger your bins the more they deviate, and the underlying function of the binomial just doesn't describe spiking processes very well: Poisson distribution works well when you know the underlying rate of events, Binomial works when you know the probability in a given interval. The former is a better description of the way people usually think about spiking neurons. – Bryan Krause Mar 27 '17 at 14:45
• @Jen Yes. If you are talking about a single bin then you have the special case of a Bernoulli distribution. If you have a sequence of bins then the two parameters for binomial are a) the probability within each bin, and b) the number of bins. The binomial distribution will describe how many of those bins are occupied (i.e., by a spike). – Bryan Krause May 12 '17 at 17:14

When trying to look at the firing of a neuron as a way to encode information, a useful trick is to model each spike as a Dirac delta function. What this means is any time point is a "spike" or "no spike." It's binary.

http://www.cns.nyu.edu/~david/handouts/poisson.pdf

It doesn't tell you anything about the internal dynamics of the neuron, that is, its electrotonic properties, but at a "macro-level" looking at spikes as delta function is a good way to look at a rate code.

A Poisson distribution gives you the probability of a given number of spikes in a span of time if you know the average rate and are independent. Because of this, you can model spike trains of "dummy" cells.

Neurons are very irregular. This has to do a lot with stochastic forces that make the interspike intervals (the time between two spikes) highly unpredictable. For convenience, a random process is used to model them. At the higher "systems" level, patterns emerge and this is how your brain makes sense of it all.

Thus, you can model on three scales:

Individual Cell, dynamical system (e.g. the Hodgkin-Huxley model) Individual Cell, rate code (e.g. a Poisson distribution model) Systems Neuroscience

https://www.tu-chemnitz.de/informatik/KI/scripts/ws0910/Neuron_Poisson.pdf