Suresh Krishna has a question about hierarchical models for neural response data:

We are interested in the neural response following the onset of a stimulus at time 0. Typically, this is a dynamic stream of (approximately Poisson distributed) impulses whose rate-parameter increases about 50 ms after stimulus onset (“latency”), rises to a peak, and then decays off over 250 ms or so to a steady state. We first isolate a single neuron and then record responses to M (say 500) repeated presentations of the same stimulus, each presentation lasting say 500 ms and separated by a 2-3 s gap. For simplicity, if necessary, we can treat responses to successive trials as “independent” of each other; the response within a trial is likely to be correlated (in a repeated-measures sense). Usually, we bin the recorded stream on a given trial into say 10 or 20 ms bins and now the measure is the number of spikes within that bin, which is commonly near-Poissson distributed. So a typical dataset would look something like:

0 1 0 1 5 5 7 8 5 7 4 4 3 3 2 ….. and so on for 1 trial and

0 0 0 0 2 5 9 8 5 4 2 3 4 2 2 …. for the second trial etc.Adding across trials produces what is called the peri-stimulus time histogram (mean PSTH), giving the expected response of that neuron to that stimulus.

Now we do the same thing for N neurons recorded in that population, and what we are really interested in is the population response to that stimulus.

Commonly, people estimate the mean PSTH for each neuron, and then just average them to construct the population response. Also, M, the number of trials recorded from each neuron is unlikely to be the same, and so there are unequal sample-size issues, so one could also take a weighted average where each neuron’s mean is weighted by its standard-error. Some others take square roots of the Poisson counts before constructing the PSTH, in order to “correct for Poisson variability”.

Now it occurred to me that this is indeed a hierarchical model, where there is an unknown distribution of PSTHs for each neuron, but let us say approximately normally distributed (or maybe a mixture of 2 or 3 normal distributions, indicating 2 or 3 neuron types) and hten within each neuron, we have some mean PSTH with each sample showing Poisson variability.

The aim is to get insight into the parameters.

Of course, the brain itself is free to pick and choose among neurons, and it probably does, and so constructing a “population” response is probably silly, but for now, but it is one of the possible decoding options that sounds fairly minimalistic in its requirements.. so it is interesting.

I’ll be meeting with him later this month, but here’s a quick neural response from me: yes, I agree that hierarchical modeling makes sense. The problem as presented is a bit too abstract for me to grab on to, but I imagine that in a particular context (a study with various treatments and replications), an appropriate Anova-type analysis will present itself.

Would the ANOVA deal with Poisson distributed data effectively? Or does this work because there are a lot of data points and/or the central (and probably most interesting points will have relatively large numbers of counts?

I am interested because we are dealing with data that have some similar features. The data are tumor numbers in older animals sacrificed at end of life. There seems to be a difference in the average number of tumors in two different lines of mice, but the numbers are relatively small (of mice and tumors) and widely distributed over ages with the possibility of some trend in average tumor number over time that may (or may not) make the numbers of tumors per animal more similar with age. Any more thoughts you have on these issues would be helpful. I could probably do the analysis in R if there are tools there, but I am still very clumsy with the software. Thanks very much.