  Identifying the right tools to express the stochastic aspects of neural
activity has proven to be one of the biggest challenges in computational
neuroscience. Even if there is no definitive answer to this issue, the most
common procedure to express this randomness is the use of stochastic models. In
accordance with the origin of variability, the sources of randomness are
classified as intrinsic or extrinsic and give rise to distinct mathematical
frameworks to track down the dynamics of the cell. While the external
variability is generally treated by the use of a Wiener process in models such
as the Integrate-and-Fire model, the internal variability is mostly expressed
via a random firing process. In this paper, we investigate how those distinct
expressions of variability can be related. To do so, we examine the probability
density functions to the corresponding stochastic models and investigate in
what way they can be mapped one to another via integral transforms. Our
theoretical findings offer a new insight view into the particular categories of
variability and it confirms that, despite their contrasting nature, the
mathematical formalization of internal and external variability are strikingly
similar.
