Mathematical Medicine and Biology Advance Access originally published online on April 20, 2006
Mathematical Medicine and Biology 2006 23(2):119-138; doi:10.1093/imammb/dql006
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Simultaneous constraints on pre- and post-synaptic cells couple cortical feature maps in a 2D geometric model of orientation preference
1 Departments of Mathematics, Biology and Cognitive Science, Case Western Reserve University, Cleveland, OH 44106, USA, 2 Departments of Mathematics and Neuroscience, Oberlin College, Oberlin, OH 44074, USA, 3 Department of Mathematics, University of Chicago,Chicago, IL 60615, USA
** Email: peter.j.thomas{at}case.edu
The most prominent feature of mammalian striate cortex (V1) is the spatial organization of response preferences for the position and orientation of elementary visual stimuli. Models for the formation of cortical maps of orientation and ‘retinotopic’ position typically rely on a combination of Hebbian or correlation-based synaptic plasticity, and constraints on the distribution of synaptic weights. We consider a simplified model of orientation and retinotopic specificity based on the geometry of the feed-forward synaptic weight distribution from an ‘unoriented’ layer of cells to a first weakly oriented layer. We model the feed-forward weight distribution as a system of planar Gaussian receptive fields each elongated in the direction matching the preferred orientation of the postsynaptic cell. Under the constraint of presynaptic weight normalization (each cell in the oriented layer receives the same net synaptic weight) and a uniform retinotopic map (displacement of centres of mass of receptive fields in the unoriented layer is strictly proportional to the displacement of the corresponding cells in the oriented layer), we find that imposing a pattern of orientation preference forces the system to violate postsynaptic weight normalization (each cell in the unoriented layer no longer sends forth the same net synaptic weight). We study this deviation from uniformity of the postsynaptic weight, and find that the deviation has a distinct form in the vicinity of the ‘pinwheel’ singularities of the orientation map. We show that uniform synaptic coverage of the unoriented layer can be restored by introducing a distortion in the retinotopic locations of the receptive fields. We calculate, to first order in the relative elongation of the receptive fields, the retinotopic distortion vector field. Both the pattern of postsynaptic weight non-uniformity and the corrective retinotopic distortion vector field fail to possess the reflection symmetry commonly assumed to relate orientation singularities with topological index ± 
. Hence, we show that ‘right-handed’ and ‘left-handed’ orientation singularities are funda-mentally distinct anatomical structures when full 2D synaptic architecture is taken into account. Finally, we predict specific patterns of retinotopic distortion that should obtain in the vicinity of ± -fold orientation singularities, if uniform pre- and post-synaptic weight constraints are strongly enforced.
Keywords: visual cortex; orientation; retinotopy; fan-in; fan-out; neural network; cortical map
Received on 24 July 2005. revised on 19 October 2005. accepted on 25 October 2005.