DECODING SECOND ORDER ISOMORPHISMS IN THE BRAIN: The case of colors and letters

We introduce a new method for decoding neural data from fMRI. It is based on two assumptions, first that neural representation is distributed over networks of neurons embeded in voxel noise and second that the stimuli can be decoded as learned relations from sets of categorical stimuli. We illustrate these principles with two types of stimuli, color (wavelength) and letters (visual shape), both of which have early visual system response, but at the same time must be learned within a given function or category (color contrast, alphabet). Key to the decoding method is reducing the stimulus cross-correlation by a matched noise voxel sample by normalizing the stimulus voxel matrix thus unmasking a highly discriminative neural profile per stimulus. Projection of this new voxel space (ROI) to a smaller set of dimensions (with e.g., non-metric Multidimensional scaling), the relational information takes a unique geometric form revealing functional relationships between sets of stimuli, defined by R. Shepard, as second-order isomorphisms (SOI). In the case of colors the SOI appears as a nearly equally spaced set of wavelengths arranged in a color wheel, with a gap between the “purples” and “reds” (consistent with the gap in the original Ekman’s color set). In the case of letters, a cluster space resulted from the decorrelated voxel neural profiles, which matched the phrase structure of the mnemonic used for more than 100 years to teach children the alphabet (across multiple languages), The Alphabet Song.

On the Linearity of Order-isomorphisms

A basic problem in the theory of partially ordered vector spaces is to characterise those cones on which every order-isomorphism is linear. We show that this is the case for every Archimedean cone that equals the inf-sup hull of the sum of its engaged extreme rays. This condition is milder than existing ones and is satisfied by, for example, the cone of positive operators in the space of bounded self-adjoint operators on a Hilbert space. We also give a general form of order-isomorphisms on the inf-sup hull of the sum of all extreme rays of the cone, which extends results of Artstein–Avidan and Slomka to infinite-dimensional partially ordered vector spaces, and prove the linearity of homogeneous order-isomorphisms in a variety of new settings.

Universal Entire Functions That Define Order Isomorphisms of Countable Real Sets

In 1895, Cantor showed that between every two countable dense real sets, there is an order isomorphism. In fact, there is always such an order isomorphism that is the restriction of a universal entire function.