  In a published paper [Sengupta, 2016], we have proposed that the brain (and
other self-organized biological and artificial systems) can be characterized
via the mathematical apparatus of a gauge theory. The picture that emerges from
this approach suggests that any biological system (from a neuron to an
organism) can be cast as resolving uncertainty about its external milieu,
either by changing its internal states or its relationship to the environment.
Using formal arguments, we have shown that a gauge theory for neuronal dynamics
-- based on approximate Bayesian inference -- has the potential to shed new
light on phenomena that have thus far eluded a formal description, such as
attention and the link between action and perception. Here, we describe the
technical apparatus that enables such a variational inference on manifolds.
Particularly, the novel contribution of this paper is an algorithm that utlizes
a Schild's ladder for parallel transport of sufficient statistics (means,
covariances, etc.) on a statistical manifold.
