As an applied mathematician, I have a growing interest in the mechanisms for uncertainty representation and computation in the human brain. In fact, I recently compiled a list of papers on this subject. But, so far I haven't found many papers which answer a fundamental sub-question:
What sources of randomness does the brain use for sampling?
Dan Goodman, a neuroscientist, suggested that I have a look into neuronal noise. The neuronal variability perspective is quite popular among computational neuroscientists and 'Bayesian inference with probabilistic population codes'(2006) is an example of a publication which demonstrates its use:
At first sight, it would seem that the high variability in the responses of cortical neurons would make it difficult to implement such optimal statistical inference in cortical circuits. We argue that, in fact, this variability implies that populations of neurons automatically represent probability distributions over the stimulus, a type of code we call probabilistic population codes. Moreover, we demonstrate that the Poissonlike variability observed in cortex reduces a broad class of Bayesian inference to simple linear combinations of populations of neural activity. These results hold for arbitrary probability distributions over the stimulus, for tuning curves of arbitrary shape and for realistic neuronal variability.
However, a recent paper published in 2018  presents compelling counter-arguments against the utility of neuronal noise for sampling in the brain.
In particular, the authors make the following case in their abstract:
Since the precise statistical properties of neural activity are important in this context, many models assume an ad-hoc source of well-behaved, explicit noise, either on the input or on the output side of single neuron dynamics, most often assuming an independent Poisson process in either case. However, these assumptions are somewhat problematic: neighboring neurons tend to share receptive fields, rendering both their input and their output correlated; at the same time, neurons are known to behave largely deterministically, as a function of their membrane potential and conductance.
While I find the arguments in  compelling due to their fundamental nature, this makes me wonder whether there are other fundamental contributions to this problem that I ignore.
Note: Cian O'Donnell recently shared his PhD thesis with me on Twitter . The neural noise vs. no neural noise variant of this question is still very much an open problem.
- Wei Ji Ma, J. Beck, P. Latham & A. Pouget. Bayesian inference with probabilistic population codes. Nature Neuroscience. 2006.
- Andre Longtin. Neuronal noise. Scholarpedia. 2013.
- D. Dold et al. Stochasticity from function - why the Bayesian brain may need no noise. Arxiv. 2018.
- R. Cannon , C. O'Donnell , M. Nolan . Stochastic Ion Channel Gating in Dendritic Neurons: Morphology Dependence and Probabilistic Synaptic Activation of Dendritic Spikes. PLOS. 2010.