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Identifying the right tools to express the stochastic aspects of neuralactivity has proven to be one of the biggest challenges in computationalneuroscience. Even if there is no definitive answer to this issue, the mostcommon procedure to express this randomness is the use of stochastic models. Inaccordance with the origin of variability, the sources of randomness areclassified as intrinsic or extrinsic and give rise to distinct mathematicalframeworks to track down the dynamics of the cell. While the externalvariability is generally treated by the use of a Wiener process in models suchas the Integrate-and-Fire model, the internal variability is mostly expressedvia a random firing process. In this paper, we investigate how those distinctexpressions of variability can be related. To do so, we examine the probabilitydensity functions to the corresponding stochastic models and investigate inwhat way they can be mapped one to another via integral transforms. Ourtheoretical findings offer a new insight view into the particular categories ofvariability and it confirms that, despite their contrasting nature, themathematical formalization of internal and external variability are strikinglysimilar.

Theoretical connections between mathematical neuronal models corresponding to different expressions of noise