MATHEMATICAL MODELLING OF THREE-DIMENSIONAL ORIENTATION PINWHEEL STRUCTURES IN VISUAL CORTEX

Orientation preference maps (OPMs) are a canonical example of cortical organization and are commonly described within a two-dimensional framework, where orientation singularities appear as point-like pinwheels. However, Nakamichi 2018 indicated that in three dimensions pinwheels rarely form straight columnar structures, challenging both the classical columnar hypothesis and existing two-dimensional descriptors. At present, a principled quantitative framework for characterizing three-dimensional OPM geometry is lacking. Here we develop a unified theoretical framework that generalizes maximum-entropy models of OPMs from two to three dimensions. In this formulation, pinwheel points naturally extend to pinwheel strings, which constitute the fundamental topological defects of three-dimensional orientation maps. Within this framework, we derive exact analytical expressions for key geometric observables, including the mean string length, the statistics of distorted string motifs, and the full distribution of orientation differences between spatial locations. We show how to apply this theory to functional [WF1] ultrasound imaging (fUSI) data from cat primary visual cortex, resolving orientation preference across cortical depth at columnar scales. The theoretical observables robustly capture experimentally observed three-dimensional organizations, revealing quasiperiodic, surface-like orientation structures and systematic distortions of pinwheel strings along the cortical depth that are not apparent in two-dimensional analyses. Together, our results provide a principled mathematical description of three-dimensional OPM organization, enabling quantitative comparisons across models and experimental modalities, and establishing extended topological defects as the natural building blocks of cortical organization beyond two dimensions.