Probabilistic tractography, Path Integrals and the Fokker Planck equation

TL;DR

This paper explores the connection between probabilistic tractography and path integrals, introducing a symmetric connectivity measure based on the Fokker-Planck equation, which improves robustness and symmetry in brain connectivity analysis.

Contribution

It establishes a formal link between probabilistic tractography and path integrals, and proposes a novel symmetric connectivity measure using the Fokker-Planck equation.

Findings

The new measure is symmetric and robust.

Proper symmetrization addresses asymmetry issues.

Experimental results validate the approach's effectiveness.

Abstract

Probabilistic tractography based on diffusion weighted MRI has become a powerful approach for quantifying structural brain connectivities. In several works the similarity of probabilistic tractography and path integrals was already pointed out. This work investigates this connection more closely. For the so called Wiener process, a Gaussian random walker, the equivalence is worked out. We identify the source of the asymmetry of usual random walkers approaches and show that there is a proper symmetrization, which leads to a new symmetric connectivity measure. To compute this measure we will use the Fokker-Planck equation, which is an equivalent representation of a Wiener process in terms of a partial differential equation. In experiments we show that the proposed approach leads a symmetric and robust connectivity measure.