David M. Alexander ^{1},
Phil Sheridan ^{2},
Paul D. Bourke ^{3}

(Introduced by Daniele Farnbach-Pralong)

^{1}
School of Biophysical Sciences and Electrical Engineering, Swinburne
University of Technology, Hawthorn 3122

^{2}
School of Information Technology, Charles Sturt University, Bathurst 2795

^{3}
Brain Dynamics Laboratory, Mental Health Research Institute of Victoria,
Parkville 3052

We provide a model of the Macaque striate cortex which predicts the
relationship between the global retinotopic mapping of the visual field and
the local receptive field mappings. We assumed that the geometry of the
striate cortex can be approximated by a hexagonal lattice, with each hexagon
representing a cortical column approximately 200um wide. An addressing
system for bijective mappings on this lattice was developed, and the address
space interpreted in terms of a Lie Algebra. Simple arithmetric operations
such as addition and multiplication were defined, and shown to correspond
to geometric operations such as translation, rotation and scaling. A unique
multiplication was shown to predict various local (i.e. columnar) receptive
field properties from the global (complex logarithmic) retinotopic mapping of
the striate cortex, The predictions of the model include:

1) general statistical
properties of receptive fields for orientation selectivity, including iso-
orientation bands, density and ratio of singularity types, and presence of
fractures;

2) position of ocular dominance bands, correct ratio between width
and length of hypercolumns, singularities arranged over centres of ocular
dominance bands, and orientation bands crossing ocular dominance borders
at right angles;

3) position of cytochrome oxidase blobs, and explanation of
their selectivity for lower spatial frequencies, contrast sensitivity
and colour;

and 4) projection of lateral connections to columns with similar receptive
fields. The model does not require the multiple free parameters found in
other models of striate receptive field maps (1). The geometric-algebraic
model has the additional advantage that it provides a functional explanation
for the architecture of the Macaque striate cortex; i.e. cortical
manipulation of visual images through rotation and scaling.