MCB111: Mathematics in Biology (Fall 2023)
week 04:
Significance: The Studentâ€™s ttest and pvalues
Preliminars
Present all your reasoning, derivations, plots and code as part of the homework. Imagine that you are writing a short paper that anyone in class should to be able to understand. If you are stuck in some point, please describe the issue and how far you got. A jupyter notebook if you are working in Python is not required, but recommended.
How many types of dopaminergic neurons?
Dopaminergic neurons are thought to be involved in reward, and they appear to signal the difference between expected and actual reward. In order to investigate this, we have recorded firing rates from optogeneticallyidentified dopamine neurons in the lateral ventral tegmental area (VTA).
One set of recordings corresponds to mouse trained in a conditional task such that the reward always matched the expected rewards: data.0. In another set of experiments, the reward was always smaller than expected: data.1

Draw histograms for the firing rate distributions for both conditions. Can you tell whether you suspect that dopaminergic neurons fire more or less under unexpected versus expected reward conditions?

How many dopaminergic neurons in data.1 appear to fire differently from those in data.0, our null hypothesis? Provide a number with a FDR (or pvalue) associated to it. How many false positives you expect to find in this selected set? Justify.
You are given two other data sets: data.2 and data.3.

Again, using data.0 as null, what can you say about each of them?

Can you draw any observation about the number of different dopaminergic neurons at play?
The role of dopaminergic neurons in reward is real see for instance this paper by the Uchida lab. The observation that there are different types of dopaminergic neurons is also documented. The data of this exercise is fake.