Modeling the Development of Ocular Dominance and Orientation Preference Maps in The Primary Visual Cortex with The Elastic Net


Andrei Cimponeriu

October 12, 1999

Work performed with Geoffrey J. Goodhill

Georgetown Institute for Cognitive and Computational Sciences,
Georgetown University Medical Center



  I. The Elastic Net model

I.1. Method: solve the Traveling Salesman Problem using an optimization approach

dsyfig1.jpg
Elastic Net development

From R. Durbin, R. Szeliski and A. Yuille, An analysis of the elastic net approach
to the Traveling Salesman Problem. Neural Computation 1, 348-358,1989.

The energy function:

energy function

{xi} - cities, {yj} - tour stops, k- decreasing parameter, "magnifying glass" with increasing magnification, requiring increased precision in getting closer to the cities.

·    first term : approximation error

·    second term : tour length

For each k - find {yj} that optimize E => approximate all tour points with the minimum length mapping. Then decrease k. Finally a solution close to the global minimum of the TSP combinatorial problem is obtained by iterative optimization.

Adapt yj according to gradient descent:

adapt weights

Repeat until k is small enough or {yj} represent a good enough approximation for {xi}.



I.2 Visual cortical maps: TSP and representing visual experience by prototype vectors

The prototype vectors to represent retinotopy (x,y), ocular dominance (OD) and orientation preference (OR):
prototypes

Prototypes represent visual experience. They can be distributed evenly (as we did) or unevenly, e.g. to model the fovea. Their spacing can emphasize one feature or another: maps corresponding to features that are assigned larger distances develops earlier and the spatial distribution of the maps is affected accordingly.


TSP (traveling salesman problem)
 V1 (primary visual cortex)
cities
      position: x, y
 prototype vectors
      features :  retinotopy = (x, y), OD = ocularity (left eye, right eye), OR = orientation preference (angle, selectivity), spatial frequency, …
the tour:
- all cities
- minimum length
 cortical activity:
- any visual input
- correlated; if correlation is given by anatomical links, it should be as short as possible
decreasing parameter k -   deterministic annealing
 k - increasing selectivity during map development(?)

Good, simple principles to build a successful model.



I.3. Dynamics of the Elastic Net

Depending on the relative "weight" of ocular dominance (OD) and orientation preference(OR)features:

comp_014_024_099_.jpg
cat-like simulation: OR map develops first


length_014_016_099_.jpg
monkey-like simulation: OD map develops first

The yj's are initialized with noisy values lying in the space spanned by the xi's and k is given a value large enough, to make all the prototypes look virtually identical. The initial decrease in E corresponds to the collapsing of the yj's to the average of the xi's. As k decreases slowly, at some point the prototypes start being distinguishable along (at least) one dimension and the corresponding map emerges. In both cases presented here, as in all our simulations, biological plausibility made us choose the x-y grid to be such that the first map that develops be the retinotopic map.


I.4. Typical OD maps and OR angle map

od_014_024_099_.jpgor_014_024_099_.jpg
"cat"

od_014_016_099_.jpgor_014_016_099_.jpg
"monkey"



II. Optical imaging and simulation maps

II.1. Experimental OD and OR maps

od_or_maps_huebener

Cat V1 (primary visual cortex). The top map is an orientation preference map; the angles are color-coded, as shown on the right. The bottom map is am ocular dominance map, where areas stimulated by one eye are white and areas stimulated by the other eye are black. Grey areas are binocular. Signals are small and also visible is shot noise from the CCD camera. Scale bar, 1 mm.

From Mark Hübener, Doron Shoham, Amiram Grinvald, and Tobias Bonhoeffer,Spatial relationships among
three columnar systems in cat area 17. J. Neurosci.17:9270-9284, 1997.


monk_od_or_obyblas_
Monkeys V1. Data from five animals. Left column: blood vessels; center column:OD maps; right column: OR maps.

From K. Obermayer and GG Blasdel, Geometryoforientation andocular dominance column
  in monkey striatecortex. J. Neurosci. 13:4114-4129, 1993



monkeys-od_
Variability in OD monkey experimental data.

From J C. Horton and D R. Hocking, Intrinsic Variability of Ocular Dominance Column
. J. Neurosci. 16(22): 7228-7339, 1996.


However, despite variability and inter-species differences, OD and OR contour maps tend to intersect at 90 degrees:

od_or_cat_cont_.jpgcat_int_ang_hist.jpg
Cat OD - OR contours and intersection angles. Left, colored iso-orientation lines and black OD contours (A) and a detail (B). Right, histograms of intersection angles between iso-orientation lines and OD borders. Black histograms are from three different cats; gray histograms are computed by overlaying an orientation map from one cat with an ocular dominance map from another cat.If the two maps come from the same animal, there is a significant bias towards90 degrees, otherwise, histograms are nearly flat.

From Mark Hübener, Doron Shoham, Amiram Grinvald,and Tobias Bonhoeffer, Spatial relationships among
three columnar systems in cat area 17. J. Neurosci. 17: 9270-9284, 1997.




monk_cont_obyblas_.jpgod_or_monk_hist_obyblas_.jpg
Monkey OD - OR contours and intersection angles. Left, contour plot of orientation preferences with the borders of ocular dominance bands. Right, histogram of angles of intersection between the two: original data a), d) and manipulated data b), c). The distribution of genuine data is significantly biased towards 90 degrees.

From K Obermayer and GG Blasdel, Geometry of orientation and ocular dominance columns
  in monkey striate cortex, J. Neurosci. 13: 4114-4129, 1993.



II.2. Simulation OR and OD maps and intersection angles


comp_014_026_099_.jpg

comp_014_022_099_.jpg

comp_014_018_099_.jpg
Simulations for different weights of the OD and OR coordinates of the prototypes. These weights cause one or the other map to develop first (after the retinotopic map) and to have a more regular appearance.

All simulation maps are qualitatively similar to experimental data. In all cases there is a bias of the intersection angles towards 90 degrees.

II.3. A very interesting case: strabismus


strabismic_cat_.jpg strabcats_fcurves_.jpg
Strabismus in experimental data. OD maps (left) and spatial frequencies (right)of normal (A) and strabismic (B) cat.
In case of strabismus OD stripes are wider and more clearly defined.spatial frequencies.
Spatial frequencies are represented as one-dimensional Fourier analyses on sections normal
to OD borders. The values near the curves are the spatial periods corresponding to the peaks.

From S Lowel, Ocular dominance column development: strabismus changes the spacing
  of adjacent columns in cat visual cortex, J. Neurosci. 14 : 7451-7468, 1994.


strabismic_sim_.jpg
"normal" / "strabismic" simulations

From G. J. Goodhill and S. Löwel, Theory meets experiment: correlated neural activity helps determine
ocular dominance column periodicity. Trends in Neuroscience, 18 (10), 1995.

Conclusions and predictions

1. Layout is a consequence of order of development

2. In cats OR maps develop first

3. In monkeys OD maps develop first



For a different approach on the same topic, see
·    F. Hoffsümmer, F. Wolf, T. Geisel, S. Löwel, and K. Schmidt.Sequential bifurcation of orientation and ocular dominance maps. In ICANN95: Proceedings of the International Conference on Artificial Neural Networks, volume I, p. 535. EC2 & Cie, Paris, 1995.
·    F. Hoffsümmer, F. Wolf, T. Geisel, S. Löwel, and K. Schmidt. Sequential bifurcation and dynamic rearrangement of columnar patterns during cortical development. In Jim Bower, editor, Computation and Neural Systems, 1996.



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