ABSTRACT
This paper presents a unified representation of the brain based on mathematical functional measures integrating the molecular and cellular scale descriptions with continuum tissue scale descriptions. We present a fine-to-coarse recipe for traversing the brain as a hierarchy of measures projecting functional description into stable empirical probability laws that unifies scale-space aggregation. The representation uses measure norms for mapping the brain across scales from different measurement technologies. Brainspace is constructed as a metric space with metric comparison between brains provided by a hierarchy of Hamiltonian geodesic flows of diffeomorphisms connecting the molecular and continuum tissue scales. The diffeomorphisms act on the brain measures via the 3D varifold action representing “copy and paste” so that basic particle quantities that are conserved biologically are combined with greater multiplicity and not geometrically distorted. Two applications are examined, the first histological and tissue scale data in the human brain for studying Alzheimer’s disease, and the second the RNA and cell signatures of dense spatial transcriptomics mapped to the meso-scales of brain atlases. The representation unifies the classical formalism of computational anatomy for representing continuum tissue scale with non-classical generalized functions appropriate for molecular particle scales.
Competing Interest Statement
Under a license agreement between AnatomyWorks LLC and the Johns Hopkins University, Dr. Michael I. Miller and the University are entitled to royalty distributions related to technology described in this work. Dr. Miller is a founder of and holds equity in AnatomyWorks LLC. This arrangement has been reviewed and approved by the Johns Hopkins University in accordance with its conflict of interest policies.