Quantitative Characterization of the Human Retinotopic Map Based on Quasiconformal Mapping

Abstract

Visual perception and cognition involve a cascade of geometric transformations from the retina to high-level cortical areas. Yet, our perception is mostly veridical and is largely invariant across individuals. The transformation from the retina to the primary visual cortex (V1) has been modeled using conformal and quasiconformal maps with the latter being a better fit to empirical retinotopic maps. However, we have not actually quantified the transformation to understand their effects in the cascade. We developed a new quantification framework for retinotopic maps based on computational conformal geometry and quasiconformal Teichmüller space theory. There are three key components: (1) pre-processing retinotopic maps to ensure that they preserve topology despite the low spatial resolution and low signal-to-noise ratio of retinotopy data, (2) quantifying the mapping between the retina and V1 (conformal or not conformal) using Beltrami coefficients, (3) developing mathematical and numerical methods to ensure the quantification is completely invertible (forward and backward transformations between visual areas). The result was a ''Beltrami coefficient map'' (BCM) that allowed us to measure and compute the forward and backward transformations between the retina and V1. We applied the new framework on the V1 retinotopic maps from the Human Connectome Project (n=181), the largest state of the art retinotopy dataset currently available. The average measurement of ''distortion'' on the left and right BCM had values of 0.310±0.110 and 0.312±0.105, respectively (range: 0 ≤ x < 1, with 0 being most conformal). We showed that the transformation from the retina to V1 is quasiconformal. Does it mean that later stages of visual processing can ''correct'' the distortions to generate veridical perception? How do visual or neural disorders affect the transformations? Future applications of this mathematical framework to all the visual areas may shed some new light on these questions.