What sources of randomness does the brain use for sampling? 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 [3] 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 [3] 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 [4]. The neural noise vs. no neural noise variant of this question is still very much an open problem. 

References:


Wei Ji Ma, J. Beck, P. Latham &amp; 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.
Not sure if I am getting your question right. You are asking for noise that has an inherently information processing function? I would add an other aspect of noises

Noise as an emerging property


On a large scale, when massive amounts of neurons are
synchronising, then you can detect Noise with Electrodes, like in
the case of EEG. EEG follows an 1/f² or 1/f³ law (brownian
noise, red noise, pink noise, depending on the literature). This
principle can be found anywhere in the world. It states that higher
frequencies / faster oscillating signals have increasingly smaller
amplitudes, similar to smaller and larger waves on the sea and so
on.
In this case is noise a byproduct of other (stochastic) processes.
On a small scale, which would probably be something you would
    measure by using 'clamping'-technique on single neurons.  The
    firing rate of a neuron (a neuron chemically charging, until it
    reaches a threshold, which turns into a all-or-nothing case) also
    follows some 'systematic randomness' on which probabilistic,
    stochastic and noisy properties might be useful to describe these
    processes.
    In this case is noise a property that emerges from natural stochastic single-neuron firing processes.


Noise emerging from signals related to certain anatomic sources


In how far I remember (I do not have a good resource on that) the
    hippocampus, which is highly relevant for memories, has a strong randomization aspect inherently in its functioning. 
    This might be my closest guess to your question?!
Another very distinct 'stochastic' process created by the
Thalamus, a largely dominant, pace-giving rhythm within the brain.


Ressource: Buszaki Rhythms of the Brain is a nice read on various of these topics

On noise as an error process

Maybe I got your question a little wrong, but thinking of the brain that in the run of evolution turned out to be highly efficient and energy saving, does not want to have any noise that does not fulfill any purpose. (But as I understand, your question was related to the functional aspect of noise.)