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

TL;DR

This paper explores the mathematical relationships between different stochastic models of neural activity, revealing that internal and external variability, despite their differences, share striking formal similarities through integral transforms.

Contribution

It provides a theoretical framework linking internal and external neural variability models using probability density functions and integral transforms.

Findings

Internal and external variability models are mathematically related.

Integral transforms can map different stochastic representations.

The formalization shows surprising similarities despite conceptual differences.

Abstract

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.